How can a red light photon be different from a blue light photon? How can photons have different energies if they have the same rest mass (zero) and same speed (speed of light)?
 A: Some areas of physics are counter-intuitive. For them, your everyday experience is a poor guide to how the universe really works. This is one of those areas.
Photons have no mass. They all have the same speed. Yet they have energy and momentum, and it isn't the same for all photons.
If you are used to $p = mv$, this doesn't make sense. The explanation is simple. $p = mv$ doesn't apply to photons. It applies to massive objects at low speeds, and photons are something different.
One way to make sense out of photons is to treat them like the new thing they are. Before you encountered quantum mechanics, you never encountered anything that was sort of like a particle and sort of like a wave. So what are the properties of this new and different thing?
An excited atom can drop to the ground state, and at the same time experience a recoil. A while later, another atom that was at rest with respect to the first atom can experience a recoil in the opposite direction and get promoted to an excited state. A photon is what happens in between. Experiments like this show that photon had enough energy to excite an atom and enough momentum to give it a recoil. They show a photon is something like a particle.
Experiments with diffraction gratings show photons have frequency and wavelength, and higher frequency/shorter wavelength corresponds to higher energies and momenta.
I am glossing over other counter-intuitive results, like uncertainty of momentum.

Having said this much, I hope I don't muddy the waters by saying there isn't any such thing as a red or blue photon. This gets back to relativity. You do have some everyday experience with Galilean relativity, which isn't entirely different from special relativity.
Suppose you are floating in space and you encounter a rock. If the rock isn't moving fast, it taps you gently. If it is moving fast, it does damage. But you can't really say how the rock is moving. You can only say how fast it is moving with respect to you. Two people could see the same rock. One could see it moving slowly, and the other fast. They would disagree on how much energy and momentum the rock has.
Suppose you are sitting in a boat watching waves go by. You count peaks passing by per second to get frequency. If you move into the waves, you encounter peaks more often, and your value for the frequency goes up. You also see the waves moving faster with respect to the boat. The distance between peaks does not change.
Photons don't have mass and their speed is always c. But their energy and momenta behave something like what you would expect from watching rocks. Their frequency behaves something like what you would expect from watching water waves or sound waves. There are differences in details, but your intuition can be something of a guide.
Photons are like rocks in that different atoms will see different energy and momenta, depending on how they move. If we repeat the exited atom experiment with atoms that are approaching each other, we find the recoil is higher than for an atom at rest, the photon has an energy higher than is needed to excite the atom. The intuitive part is that the photon "hits harder" when you run upstream into it. The counter intuitive part is that photons always travel at c, so it hits at the same speed.
You also get semi-sensible results when an atom and a diffraction grating are approaching each other. Like water waves, the diffraction grating encounters peaks more often and sees a higher frequency. The counter-intuitive part is that the speed doesn't change, but the distance between peaks gets shorter. The diffraction grating reflects the photons at a different angle.
So there is no such thing as a red or blue photon because it matters how fast the thing it hits is moving. The thing it hits will see it as red or blue, and something else would see it differently. But again, this is counter-intuitive. Even though the photon always hits a speed c, there is a difference. It is more intuitive when you think of the relative velocity between the thing that was hit and the thing that emitted the photon.
Quantum mechanics is often like this. There are two interactions, and everything adds up before and after. But what goes on in between can be murky. A photon or electron is emitted from a source. There is no trajectory it follows, only a wave that describes probabilities. Then it hits something. The recoil of the source and target match.
Intuition has lead people to look for a deeper theory that explains more. If there is a cause, there must be a predictable effect. It turns out that this intuition leads down a wrong path. This is the way the universe works. The best thing to do is find ways to get used to it.
A: Let me add a few things.


*

*A photon is an elementary particle, and as long as it propagates, it is in a superposition of states, meaning it is in a superposition of frequencies, and does not have a well defined frequency. You cannot know its frequency until you interact with it or absorb it.



As a quantum mechanical entity photons can be in superposition

Does a single white photon exist?


*A photon, as long as it propagates, could be viewed from different reference frames, and since there is no universal reference frame, the red wavelength photon could be viewed as blue from another reference frame. You cannot know its frequency until you interact with it or absorb it.


Why does the motion of the emitter (doppler shift) impact the energy of the photons


*Let's say you emit a blue wavelength photon, and it travels in expanding space, and undergoes cosmological redshift. The absorber will see it as a red wavelength photon. Who is right, would you call that a blue or a red wavelength photon?


https://en.wikipedia.org/wiki/Redshift
A: This question is still yet fully defined in Physics, because they require the analysis of light to be a duality; in which they are understood as both:

1) a “particle”, called a photon.
2) a “massless wave”, measured by it’s frequency. I believe the theoretical issue lies between:
A) ”Newtonian Physics” (rules governing our understanding of physics larger than the atomic level); this set of rules accurately describes “Fluid Dynamic” (rules governing our understanding of the physics of fluids and gases) and “Thermal Dynamic” (rules governing our understanding of the physics of heat exchange and molecular combustion). 
B)”Electrodynamics” (rules governing our understanding of physics of atomic and electromagnetic energies), which did not seem to follow these same theoretical rules. 

The bridge of these two fields, I believe, is found in the rules of “General Relativity” (rules governing physics of “Matter” traveling slower than the speed of light) and “Special Relativity” (rules governing physics at the speed of light and/or with no “Mass”).
When we discuss the features of “light” in terms of color, we observe the wave frequency of the light ray. In this analysis, we do not incorporate the Matter of a photon as a “particle”. Rather, we analyze its “energy output” as a wave with a particular frequency capable of transmitting through a vacuum (thereby we understand it does not require any Matter component, by which we negate Mass). 
The chart below shows the different energy frequencies of both the visible spectrum and the larger electromagnetic scale from radiation to radio waves.
￼
The ongoing question is that we can observe that those electromagnetic and light-speed phenomena are effected by various physical objects and, albeit in the most extreme circumstances, “The Weak Force” (Gravity). Since we observe this, we suppose that light and energy have features that would imply a “tangible particle” traveling on a path. Thus we currently treat light theoretically as a “particle” and a “wave” simultaneously while it would seem incongruous. To my knowledge, the “Matter composition” of a photon is yet to be established beyond some of its observed characteristics. I believe this is one of the current issues at heart of the unestablished theory to align “Quantum Mechanics” (the rules governing physics at the subatomic level, of which a photon particle would be classified as) and “General Relativity”.
A: They differ in their energy. Special relativity states that $E=\sqrt{m^2c^4 + p^2c^2}$. For a massive particle, there is a one on one relation between its energy and speed. In the limit $m \rightarrow 0$ this is no longer the case. All massless particles move at light speed, but their energy/momentum can vary.
A: The only difference between the two is the energy they have. 
$$
E=\frac{hc}{\lambda}
$$
As you can see from the equation above, different energies means different wavelengths. Different wavelengths means different colors. 
It is important to know that even though photons are always massless and always move with the speed of light, that does not mean that they always have the same energies as can be seen from the equation above.
A: "different color" is a feeling in your brain. Red and blue is different feeling, the root of the different feeling is some different property of the photon that can result in different feelings. 
In the case of human eye, the property that make the difference of feeling is the frequency/energy of the photon. Photons with different energy stimulate light sensors in retina with different strength. Blue photons stimulates blue sensors more, red photons stimulates red sensors more, finally giving different feeling of colors in your brain.
At very low light condition like at night, a fourth type of light sensor that responses to differnt visible light photons not very differently is stimulated much more then blue and red sensors, then most the visual signal sent to your brain is from that fourth type of light sensor and this is why you can't see color well anymore at low light.
All this is for human eyes. Other eyes (including bio engineered eyes) can have differnt types of color sensors and even not generating signal to brain based on the frequecy/evergy property but on other properties, like polarizing?
