Is the many-worlds interpretation really just an interpretation? Is the many-worlds interpretation just a different interpretation to quantum mechanics or does it contain some different predictions?
In other words, is it possible theoretically to conduct an experiment that checks the many-worlds interpretation?
 A: Many worlds interpretation and De Broglie-Bohm theory have pretty much the same predictions.
Copenhagen interpretation is actually under defined since it is not scientifically specified under what physical circumstances does the wavefunction collapse. Under certain specifications of the collapse there are experiments that in principle could distinguish Copenhagen from many worlds and De Broglie-Bohm theory.
Spontaneous collapse models (similar to Copenhagen except the physical conditions for collapse are now rigorously described) make predictions different than those of many worlds.
A: A theory is a proposition of scientific fact, an interpretation is one of conceptual underpinnings. Whether a given variant of quantum mechanics is a theory or an interpretation depends on whether or not it can in principle be put to experimental test, variously described as verification or falsification according to your level of pedantry. This hinges on whether or not its predictions differ from those of standard QM.
Proponents of many-worlds like to believe that it is in principle falsifiable and that one day they will wring some useful predictions out of it (much as Bell did for nonlocal quantum entanglement).
Opponents of many-worlds see that as a pseudoscientific hiding to nothing, on a par with waiting for aliens to land a flying saucer on the White House lawn. They point out that the original motivation for the theory was to tighten up the conceptual foundation for mainstream predictions; if it had predicted anything different it would have been deemed to have failed. Its whole rationale was to be made unfalsifiable. And as of now, that is exactly what it is. Yet somehow, that has been transformed into a wistful search for falsifiable "black swan" predictions, a clear mark of pseudoscience.
Opponents also point to deep problems with conservation laws, the nature of consciousness and self, and suchlike.
Mainstream physics is not waiting up; unless and until many-worlds achieves a Bell-like breakthrough, it is just fantasising.
A: Build a  Elitzur–Vaidman bomb tester.
Make the human being in a box the bomb, and have many many photons.
The photons that enter the box and hit the human take a picture of what the human is doing.
We set up the box so that we can reverse time -- do a unitarily reversal of everything in it -- if no photons hit anything in the box.
The human will either be in one spot (where the photons take their picture), or not (where the photons would miss the human).
We then put the human in the box.  They either become armed (stand where the photons would hit them and take their picture) or not (stand somewhere else).
We then reverse time (unitarial reversal of everything in the box) if they aren't hit by photons.
The box opens.
If we reversed them, they come out in exactly the same state they where in when they entered the box.  No memory of what happened in the box.
If we didn't reverse them, they come out with a memory of what happened in the box.
"Bomb exploded" is "human in box has a memory of what happened in the box, and was hit by photons".
"Bomb did not explode" is "human in box has no memory of what happened in the box".
The Photons we fire can draw a picture of the human doing what the human has no memory of doing, basically a photo from another branch of reality.
The real fun thing is, barring something I don't understand, we can use the EV bomb tester to send that human a message with those photons (timing of when we send the photons, say), and have them reply to our message by moving around (if the first photons didn't "set off the bomb", we can do it again).  We can have a conversation with not only that human, but multiple different humans in superimposition (maybe something stops this, but I don't see it?).  Then we can back them up to the human who didn't experience any of it.
This is insanely technically infeasible.  And I'm talking about insanely insanely.  It makes detecting graviton particles look easy, and the easy part of detecting gravitons is getting a Jupiter sized detector in close orbit of a neutron star and then filtering out every neutrino interaction without the neutrino shield forming a black hole.
I'm also uncertain if you could shove a quantum random number generator into the box with the human, have the human act differently depending on the number generated, and get a picture of the superimposition of the actions via this method.  Does the photon that doesn't enter the bomb box interfere with that?

The trick here is that the EV bomb-tester lets you take pictures of "an erased branch"; which means (if I'm right) you can take pictures of multiple erased branches.
If we can have a box in which multiple different "branches" of classical physics occur with humans acting differently in it, then erase them back to their initial state (at a quantum mechanical level; ie, perfectly), and then export pictures of what happened on the branches (heck, the humans on the branches can pass the Turing test), it is pretty hard to argue that the inside of the box didn't experience multiple worlds.
Of course, the "time reverser" engine is a bit like the chinese room "machines cannot be intelligent" trick; the machine required to be able to it breaks everyone's intuition.
A: Both CI and MWI are approximations to a complete quantum-mechanical description of the measurement process. These are generally excellent approximations in the sense that we will never be able to observe wave interference between a dead cat and a live cat, or between an experimenter who saw X and her alternate self who saw Y. The accuracy of these approximations depends on the macroscopic size of things like cats and experimenters. For mesoscopic systems, these approximations are poorer. Here is a nice paper that simulates measurement by a mesoscopic system:
Allahverdyan, Balian, and Nieuwenhuizen, "A sub-ensemble theory of ideal quantum measurement processes," 2017, https://arxiv.org/abs/1303.7257
They see a variety of phenomena that cannot be explained in the Copenhagen or many-worlds approximation, such as a variety of time scales, none of which is present in Copenhagen or many-worlds.
A: I'm adding a second answer to clearly demonstrate two experiments that could, in principle, be performed which would give different results under the Copenhagen or Everettian (Many Worlds) theories. The first experiment is a Wigner's friend experiment and the second is a double slit experiment with a humans in a Hamster balls (credit for this latter experiment to a conversation with Jess Reidel).
Wigner's Friend
In the Wigner's friend experiment Wigner has a friend who goes into a room. The friend then prepares a qubit in a superposition state $\frac{1}{\sqrt{2}}(|0\rangle + |1\rangle)$. The friend then measures the qubit. On the Copenhagen theory the state of the system collapses as soon as the friend performs the measurement. On the Everettian theory the friend becomes entangled with the qubit and the total state inside the room including the friend and qubit is an entangled superposition state. Wigner then opens the door and asks his friend what she observed.
Wigner's Friend with Qubits/Qutrits
To butter you up for the full Wigner's friend experiment first I'm going to describe the case in which Wigner's friend is just a qutrit, a three level system which will interact with the qubit. The three states are $|R\rangle$, which means the qutrit is "ready" to interact with the qubit, and $|0\rangle$ and $|1\rangle$.
The qubit starts in the state $|0\rangle$. It then undergoes a Hadamard gate to bring it to a superposition state. The Hadamard gate is given by
$$
U_H = \frac{1}{\sqrt{2}}\begin{pmatrix}
1 & 1\\
1 & -1
\end{pmatrix}
$$
The top (left) row (column) corresponds to qubit state $|0\rangle$ ($|1\rangle$). Note that $\hat{U}_H$ is it's own inverse.
The qubit and qutrit then now undergo an interaction described by unitary $\hat{U}_I$ which has the following effect on the joint system:
\begin{align}
|0R\rangle \rightarrow_{U_I} |00\rangle\\
|1R\rangle \rightarrow_{U_I} |11\rangle\\
\end{align}
An example of a unitary that could effect this transformation is
$$
U_I = \begin{pmatrix}
0 & 0 & 0 & 0 & 1 & 0\\
0 & 0 & 0 & 1 & 0 & 0\\
0 & 0 & 1 & 0 & 0 & 0\\
0 & 1 & 0 & 0 & 0 & 0\\
1 & 0 & 0 & 0 & 0 & 0\\
0 & 0 & 0 & 0 & 0 & 1
\end{pmatrix}
$$
The rows (columns) correspond to the states $|0R\rangle, |1R\rangle, |01\rangle, |11\rangle, |00\rangle, |01\rangle$. The reader can confirm that this matrix is unitary and that it effects the transformation above. In fact, $U_I^{-1} = U_I^{\dagger} = U_I$.
This unitary acts on the joint initial state to give
$$
\hat{U} \frac{1}{\sqrt{2}} (|0\rangle + |1\rangle) \otimes |r\rangle = \frac{1}{\sqrt{2}} (|00\rangle + |11\rangle)
$$
The qubit and qutrit are in an entangled state. The qutrit is in $|0\rangle$ ($|1\rangle$) if the qubit is in $|0\rangle$ ($|1\rangle$).
When Wigner examines the qubit and the qutrit half the time he sees the qubit in $|0\rangle$ and the qutrit in $|0\rangle$ and the other half of the time he see the qubit in $|1\rangle$ and the qutrit in $|1\rangle$.
But now Wigner does a different experiment. First the qubit undergoes $\hat{U}_H$, then the qubit and qutrit undergo $\hat{U}_I$. But now, the qubit and qutrit undergo $\hat{U}_I^{-1}$ and the qubit undergoes $\hat{U}_H^{-1}$. This essentially reverses the previous dynamics. The state is reverted back to
\begin{align}
|0R\rangle
\end{align}
For this second experiment, every time Wigner looks at the qubit and qutrit he sees the qubit to be in state $|0\rangle$ and the qutrit to be in state $|R\rangle$.
Wigner's Friend is a Human
The Wigner's friend experiment is the exact same as above with the qutrit replaced by a human. The human has 3 effective states, some initial state $|R\rangle$, the state of having seen the qubit as 0: $|0\rangle$ and the state having seen the qubit as 1: $|1\rangle$. The "observation" the qubit by Wigner's friend is enacted by the unitary matrix $U_I$ describing the dynamics inside the room while Wigner's friend examines the qubit.
For the first experiment described above, we have the classic Wigner's friend experiment.
On the Everettian theory the final state of Wigner's friend and the qubit is
$$
\frac{1}{\sqrt{2}}(|11\rangle + |00\rangle)
$$
When Wigner opens the door, half the time he finds the qubit in $|0\rangle$ and his friend having observed the qubit in $|0\rangle$ and the other half he finds the qubit in $1\rangle$ and his friend having observed the qubit in $|1\rangle$. On this theory, we maintain that Wigner's friend was indeed in a macroscopic superposition of quantum states. An idea which is unsettling to some.
On the Copenhagen theory, if we put the Heisenberg cut below Wigner's friend's observation, then we have that Wigner's friend CANNOT be in a superposition state, and that the wavefunction collapses when Wigner's friend measures the qubit. In this case, 50% of the time Wigner's friend is in the state $|00\rangle$ along with the qubit and $|11\rangle$ the other 50% of the time.
If we put the Heisenberg cut below Wigner's observation, but above Wigner's friend's observation, then, just like in the Everettian theory, Wigner's friend enters an entangled superposition state with the qubit and the quantum state only collapses when Wigner opens the room. This approach raises the major philosophical question of "what makes Wigner special that the wavefunction collapses when HE makes an observation but not when his friend makes an observation?"
All of this is fine. The 3 theories I have described (Everettian, Copenhagen 1 (low Heisenberg cut) and Copenhagen 2 (high Heisenberg cut)) all give the same prediction at the end of the day for what Wigner sees.
But now Wigner performs the second experiment in which the unitaries are time-reversed before his final measurement.
It is clear that on the Everettian theory (and the Copenhagen 2 theory) that Wigner will get the same exact results as he got with the qubit and qutrit. He will ALWAYS observe the qubit to be in state $|0\rangle$ at the end of the experiment. He will also always observe his friend to be back in the ready state.
But what would happen on the Copenhagen 1 theory? After $\hat{U}_H$ and $\hat{U}_I$ the system would be in state
$$
\frac{1}{\sqrt{2}}(|00\rangle + |11\rangle$
$$
But, because the system collapses during Wigner's friends measurement. The system is EITHER in $|00\rangle$ OR $|11\rangle$. Suppose the system is in $|00\rangle$. Now when the system undergoes $\hat{U}_I^{-1}$ the resultant state will be
$$
\hat{U}_I^{-1}|00\rangle = |0R\rangle.
$$
This is fine and makes sense. But when the inverse Hadamard acts on the qubit the final state is
$$
\hat{U}_H^{-1}|0R\rangle = \frac{1}{\sqrt{2}}(|0\rangle + |1\rangle)\otimes|R\rangle
$$
For the case where the state collapsed to $|11\rangle$ after Wigner's Friends measurment the final state would have been
$$
\frac{1}{\sqrt{2}}(|0\rangle - |1\rangle)\otimes |R\rangle
$$
In this case Wigner will always see his friend in the state $|R\rangle$, but now 50% of the time he will find the qubit in state $|0\rangle$ and 50% of the time he will find the qubit in $|1\rangle$.
**Therefore Copenhagen 1 theory makes DIFFERENT PREDICTIONS about this experiment than the Everettian and Copenhagen 2 theories.
Double Slit - Humans in a Hamster Balls
Researches have performed experiments in which particles with dozens of atoms have exhibited double slit interference. Suppose we could theoretically continue scaling this experiment up to large and large systems. Until eventually, we can perform an experiment in which we can put a human in a steel ball, shoot them through a double slit experiment, and look at where on a screen on the other side the human crashes.
On the Everettian theory as the person traverses the apparatus they would go into a superposition state of taking one path versus the other. So, mid-flight, they would be in a superposition state. Their paths would then interfere at the screen and, if we shoots 1000's or hundreds of poor people through this apparatus, we would see an interference pattern build up.
On the Copenhagen theory it would be forbidden for the human to be in a superposition state because they are a macroscopic system. Instead, after passing through the apparatus, they would probabilistically take one path or the other and no interference pattern would emerge. Just 2 blobs like in the double-slit experiment when it is known through which slit the particle passed.
Again we see a discrepancy between the Everettian and Copenhagen theories. If someone wants to defend the Copenhagen collapse theory my opinion is that the onus is on them to explain at what point, between dozens of atoms, and a full human, does it become fundamentally (as opposed to technically) impossible to perform a double slit experiment and why. This latter effort is exactly what is done by researchers looking into spontaneous collapse theories and non-linear extensions to the Schrodinger equation. I respect this effort. The bare Copenhagen theory, however, does not give any rigorous physical explanation for where the Heisenberg cut appears and why. And for this reason I claim the Copenhagen theory is incomplete

A note on terminology: Here I have (infinitely begrudgingly) used the convention that two interpretations make the same physical predictions. I've done this so I don't have to fuss about the terminology with folks who insist on this definition of interpretation. My preferred definition of "interpretation" is that an interpretation of quantum mechanics is any quantum theory which attempts to address the measurement problem using some approach. Anyways, for this reason you will see that I didn't use the word "interpretation" anywhere above, because I am, of course, discussing the possibility that the Copenhagen and Everettian stories make different predictions.
A: Despite the popular belief, the many-worlds formulation does make at least one testable prediction, at least from the first-person perspective, that is different from the standard Copenhagen interpretation, IMO. Full disclosure, my coffers are filled with the inter-world tax-evasion money of the many-worlds mafia.
Of course, it is the famous quantum suicide experiment. You hook the trigger of your gun to a measuring device that measures the $z-$component of the spin of a spin$-\frac{1}{2}$ particle. You set up the apparatus such that if the measurement of the spin comes out to be up then the gun fires, and otherwise, it does not. You prepare a spin$-\frac{1}{2}$ particle in an eigenstate of the spin in $x$ direction, and set the timer on the measuring device to measure the spin in $2$ minutes. Put your head in front of the gun, take a really good sleeping pill (and an anti-dote) such that it will put you in a coma for precisely $4$ minutes and the anti-dote will wake you up at the end of it.
According to the many-worlds formulation, there is always one branch of the multiverse where "you" will experience waking up because there is always one branch of the multiverse where the measuring device measured spin down and didn't fire; and in the branch where it did fire, you will die peacefully without knowing it because you are in a coma (I hope that's how comas work). Thus, in the history of that branch, you survive all the experiments.
Whereas if you put your friend in front of the gun, you will see that their head gets blown in around half of the experiments (just like yours did in all those other branches without your knowledge).
Now, one can obviously say that this shows nothing because the Copenhagen interpretation also predicts a $1/2^n$ chance that you are alive at the end of $n$ experiments. However, if you were to see that roughly $500$ of your friends' heads get blown if you do this experiment on $1000$ of your friends but you survive all the $1000$ times when you do it $1000$ times on yourself, you would be hardpressed to change your mind and put very high credence in the many-worlds formulation.
Of course, as I said, this is a first-person test. You can make it so that there are two guns tied to the same measuring device and then both you and your friend can become sure of the truth of the many-worlds formulation.
A: "Interpretation" should be used only when , after calculating specific distributions to compare with measurement, the values are the same, so one cannot decide for one mathematical model versus the other.
As I understand the many worlds "interpretation" started from accepting the path integral formulation postulating that all possible paths exist. As the path integral formulation gives the same calculated results as the usual  field theoretical calculations  based on the postulates of quantum mechanics, it is an interpretation. Note that there is no "collapse postulate" in the postulates. Just that measurements can check the probability distribution given by the calculations (the wavefunction postulate).
In general, theories in physics have a range of the values of variables where their predictions are valid within experimental errors with data. Newtonian physics breaks down both in quantum dimensions and in large dimensions, for example.
Bohm's pilot theory is an interpretation  for non relativistic energies, and people working on it are struggling with the relativistic case. The way the "many worlds" is taken up in popular science, is way off interpretation. It is an independent  theory, and the other answers discuss the possibility  of checking it with measurements.
A: If you could design a decoherence-free box, put someone inside it, have that person measure a qubit, and then unitarily reverse the whole thing to before measurement was ever done, it would look extremely bad for Copenhagen, and extremely good for many-worlds. I can't imagine anyone reasonable would still believe in wavefunction collapse after that. However, it is absurdly unlikely (to say the least) that anyone will ever manage to perform such an experiment, so it boils down to how large/complicated/massive a system do you need to see in coherent superposition before you give up wavefunction collapse.
Edit: Let me clarify the thought experiment due to discussions in the comments. Imagine there is a Copenhagen verifier, a Many-Worlds prover, and a neutral third party. The Many-Worlds prover aims to prove to the Copenhagen verifier that Many-Worlds is true. The Copenhagen verifier prepares a qubit in a state unknown to the Many-Worlds prover, and it is put inside the decoherence-free box (the mechanism of which is known and trusted by the verifier). A neutral third party, which both of them trust, is placed inside the box, and once the experiment begins, given enough time to measure the qubit (you could also make it so that an automatic mechanism measures the qubit, and shows the result to the person, or whatever). The Many-Worlds prover then (somehow) unitarily reverses the whole thing to before measurement was done. The Copenhagen verifier can then take the qubit and verify it is indeed in the state they prepared it in. (Naturally to accomplish this with high certainty the procedure will have to be repeated many times.) If all this occurs successfully, the notion that the state of the qubit collapsed upon measurement becomes untenable in my view.
A: An interpretation is a mapping from the formalism of QM to the real world. It does not, by definition, disagree with the predictions of QM at all.
Many world believes that the various possible outcomes of experiments actually exist, Bohm claims that particles are "real" and move along complex paths, Copenhagen says that QM is the final word and you can't go any deeper, etc. Experiments based on Bell's Inequality only demonstrate QM behaviour that is counter-intuitive, interpretations still have to "explain" the behaviour.
So the answer to your question is No, it is NOT possible to conduct an experiment that checks whether the many worlds interpretation is correct. But you might be able to find an QM behaviour that is incredibly hard for other interpretations to explain.
Please note that "interpretations" are not interpretations if they challenge any of the predictions of standard QM.
A: Actually, there is one experiment documented that could just prove this. It is based on proving that communication exist between the proposed parallel worlds. They basically isolate a particle in an ion trap. Then they make a quantum measurement on another system (with two discrete outcomes), creating two parallel worlds (branching). Depending on the measurement, the ion is excited (before it decoheres) only from one of the parallel worlds. A detection of this excitation is evidence for the MWI.
The key to this experiment is time and decoherence. The ion must be trapped long enough so that the effects from the parallel worlds can affect it before decoherence.

I will show in section 4 that these single ions are isolated from the environment
to such a degree that the decoherence timescale is on the order of seconds or longer with
existing technical ion-trap equipment. Moreover it is possible to excite these atoms before
they are correlated with the environment to such a degree that complete decoherence took
place. In our example above Silvia1 switches on the microwave emitter long enough to
excite an ion in a trap with a large probability. After that, Silvia2 measures the state
of the ion and finds that it is excited with some finite probability, though Silvia verified
it was in the ground state before the branching took place. From that Silvia2 infers the
existence of Silvia1. In an obvious way Silvia1 and 2 can exchange informations (bit strings
of arbitrary length), e.g. by preparing more than one isolated ion. Single ions in traps can
act as \gateway states" and communication between parallel worlds is possible.

https://cds.cern.ch/record/289177/files/9510007.pdf
So the ion must be isolated from the environment long enough (because any information leak to the environment causes decoherence) so that they can detect any effects from the parallel worlds.
So the answer to your question is that yes, theoretically it is possible to conduct an experiment. The MWI is a unique one, because it is the only one that actually deals with many existing worlds so if this could be proved then it would change the way we think about QM.
