Many-worlds interpretation vs 'just' randomness? I have this question about MWI I always wanted to ask but never dared to! It could be that I just don't know enough physics to understand the answer, or the question!
Anyway, here goes: What is it that differentiates MWI from a physical theory (let's call it 'pure random') which is equivalent but where, in every instance where MWI asserts that the universe spawns/splits up (Schrödinger's cat-esque situations), this theory simply asserts that the universe flips a coin and randomly1 decides which of the possible universes should be realized. In this theory, the other forks are never realized, so there's always just one universe in one state making a bunch of random choices how to proceed.
To me this theory seems less objectionable than a theory postulating an enormous amount of parallel universes that can't see each other anyway. So my question is, is there any physical reason to prefer MWI over the 'pure random' theory (and what is this 'pure random' theory called anyway)? After all, the parallel universes in MWI are inaccessible anyway, so it doesn't seem to have any consequence they exist. It's just another way of expressing randomness... and it doesn't even seem MWI can fully do away with randomness anyway. (After all, why am I experiencing this particular universe, and not one the many other ones MWI asserts exists? To me it seems I'm experiencing a random one among the possible ones.)
I'm wondering if my 'pure random' theory is really the same thing as what MWI-believers talk about, albeit they use a more colorful language and that's why it gets presented this way in the popular press. But on the other hand many presentations of MWI (including Wikipedia) seem to imply these multiple universes really do exist, and that Everett was of this belief. 
Apart from the obvious consequences to things like quantum immortality (which Everett also believed in, further underlying he believed in the existence of these parallel universes), are there other observable/measurable differences between MWI and pure random or other reasons to prefer this?

1 The probability distribution would be to reflect the event in question, just as I imagine that MWI would in some situations split up several universes with various properties, and the 'concentration' of a given property reflects the probability of measuring that property... I hope it makes sense!
 A: The question uses the word "theory" repeatedly, but interpretations of quantum mechanics aren't physical theories. There is no testable prediction of the MWI, for example, that would allow us to falsify it (except in the trivial sense that we could falsify quantum mechanics in general, which would also falsify the Copenhagen interpretation and every other interpretation).
So, if there are reasons to like or dislike your interpretation, they would be philosophical reasons, not empirically meaningful ones. One possible reason to dislike it is that quantum mechanics is unitary, but your interpretation introduces a nonunitary process of randomly pruning branches off of the universe. This is also an objection that applies to the Copenhagen interpretation (CI), and the defense against it would probably be the same: that the nonunitary process has no observable effects.
Another objection would be that although we often visualize a cartoon version of MWI in which the universe splits at discrete points, like a tree, in fact this isn't what MWI describes. MWI describes a smooth process. Therefore it's not clear to me how we would define this process of "pruning," since there's no "tree" in the first place.

But in case there is - by definition - no physical difference between interpretations, then my question is more like, why do so many people prefer MWI over pure random.

Do you mean why do so many people prefer MWI over CI?
A problem with CI is that it talks about measurement and observers, but measurement is just a physical process that should go according to QM, and observers are just systems of particles that should interact according to QM. Therefore it's odd that only observers have the ability to trigger wavefunction collapse. MWI doesn't have this problem.
A: It's important to understand that the size of the state space in the many worlds or Everett theory is constant:  it doesn't increase when the individual "worlds" or eigenstates "split up" or become macroscopically distinguishable.  A consequence is that just as the "worlds" can "split up", they can also "merge", such as when the two spin eigenstates of an electron are separated, and then come back together.  Experimentally, these two eigenstates interfere with each other when they come back together - in some sense they "know" they are two versions of the same electron.
This brings up the question as to whether your "pure random" theory is actually equivalent to Everett theory.  When your theory "flips a coin" and follows the "heads" result, is there an invisible "tails" result that also proceeds and, if it encounters the "heads" result, causes the "coin" to be "unflipped"?
If no - that is, once you "flip the coin", you forget about the possible result that the coin doesn't choose - then your theory is not equivalent to Everett theory, and in fact is wrong:  the theory cannot explain observed experimental results in some quantum mechanics experiments.
If yes - that is, your theory has to keep track of the nonchosen options, and know when to "unflip" the coin, then your theory becomes just as complex as Everett theory, since just as Everett theory has to worry about all those other worlds, your theory has to worry about all those nonchosen options.
A: MWI doesn't posit anything beyond what's already contained in the math of ordinary quantum mechanics: the novel idea of MWI was a different approach to analyzing the state.
Superficially, the qualitative behavior of quantum mechanics drastically contradicts the seemingly classical macroscopic universe we're used to seeing every day. So, initially, quantum mechanics was interpreted as a theory that was only applicable to microscopic scales, and when you have clearly delineated microscopic and macroscopic problems, one can invoke wavefunction collapse as a hack connecting the two that gives the correct experimental results. A true theory of everything could be neither quantum nor classical, but have to somehow interpolate the two.
Everett's observation is that relative states — i.e. looking at the state space of subsystems — has the basic mathematical features of classical mechanics that had previously been thought to be completely incompatible with quantum mechanics.
This observation reopens the possibility of interpreting quantum mechanics as a true theory of everything.
The description in terms of "splitting (or recombining!) parallel universes" only truly happens in subsystems, and the reason splitting is taken seriously is because the way probability theory is introduced does not admit an 'ignorance interpretation' as it does when applied to classical mechanics: all of the different 'worlds' as seen in the state of the subsystem could potentially still interfere with one another depending on how they're entangled with the rest of the universe.
Also, the splitting isn't even a uniquely described thing  — e.g. a totally mixed qubit can be equally well described as "half spin up half spin down around the $z$ axis" or "half spin up, half spin down around the $x$ axis", or even "25% spin up 25% spin down around $x$ and 25% spin up 25% spin down around the $z$ axis" or more exotic things.
