Is the wave nature of electron just its probability wave or something more? Light is said to act like both wave and particle. When we talk about an electron we say it's a particle and the wave portion is just its probability and not a physical wave like light.
So is there some actual wave nature of the electron itself like light, or is it a wave due to its probability only?
If that is the case then doesn't the symmetry kind of break?
 A: Light is a particle in that, its energy is quantized.
The quanta of radiation are called: photons.
An electron is a wave in that, its "collective"[ensemblic] behavior is that of waves( double-slit experiment for particles).
Light wave-particle duality is solved in the quantum field of light.
Indeed, a quantum field of light is a probabilistic wave, where for example the phase is undetermined and the domain is only localized.
When one deals with the quantum fields this disparity you referred to is minimized if not formally overcome. Light becomes a rather probabilistic wave or more precisely an indeterministic physical wave.
In fact, it was Einstein who showed that in order to justify the particle-like contribution to the vacuum energy fluctuations of the black body radiation(apart from the (physical)wave-like vacuum fluctuations), one should think of light as a probabilistic(quantum) wave.
How do you visualize a quantized electromagnetic field?
A: The most we can say for certain is that the wave function associated with an electron allows us to determine the probability of finding the electron at a given point in space. All the experimental evidence suggest that electrons are point like particles, so most physicists would not agree with you if you supposed that the wave function was actually the electron in a smeared out state. Beyond that, there is little we can say about what the wave function 'really' is; there are different interpretations of quantum mechanics, but they are just that- interpretations.
A: Physics isn't about reality, it's about what we can measure.
When we detect light, it's usually from its effects on atoms. A photographic film has crystals, and a few photons can be enough to get a whole crystal to change state. Things like that. Our measurements are always quantized, it's always a whole crystal that changes state or doesn't change state.
Similarly, when atoms change state by emitting light, they always emit a quantized amount of light.
Light appears to travel in the form of waves. But atoms absorb and emit photons which can be assumed to not act like waves.
It's easy to explain diffraction with waves. If light goes through a slit that's 3 wavelengths wide, and it travels in all directions, there will be an angle where the light from three different peaks all arrives at the same time because the light from different parts of the slit travels different distances to get there. If you have a detector that depends on some amplitude of light, it gets three waves cancelling. At a different angle it's light from just 2.5 different waves that arrive at the same time, and so just half a wavelength doesn't cancel. You see 1/5 of the light. It all makes sense.
Your detectors might take some time to detect. A weak light wave might take longer to build up enough energy to change a crystal on the film. Or maybe the crystals change state, and sometimes they are very sensitive and other times less so. A very few might change quickly from a low-amplitude wave, while others are less likely to do that.
But if atoms emit whole photons that each travel until it is entirely absorbed by one other atom, how can you have diffraction? Each photon is either absorbed or not absorbed, independent of any other photon. Diffraction can only be a probability of the photon taking each particular path. If we assume that each individual photon has its own identity as it travels, that's much harder to visualize. But we know it's absolutely true because it is the only possible way to interpret the quantized measurements. ;)
Similarly with electrons. We know that electric charges on tiny oil drops are quantized. Charge is always quantized. Each electron has a unit charge, and a charge and an electron both have to be someplace specific and can't be smeared out across a wave.
Now imagine that a traveling electron changes its state at some rate. For example, it might be spinning, or suffer some other periodic state change. Then if you send a lot of electrons through a slit, and they spread out, there is an angle where electrons with three periods will all arrive at once. And if your detector only detects electrons that are in the right state, they will cancel. The part that doesn't cancel might change the state of the detector so that it somehow stores a potential until it has enough to record a detection. So we would get diffraction. All that's needed is that the electron has a periodic change of state, and the detector detects the sum of electron states at one location. (And maybe that it can store a quantum of electron energy over time.) The electrons could simulate a wave. Or they could in fact travel as waves.
But let's get past all this hypothetical theorizing.  All of our measurements are quantized. We know beyond the shadow of any doubt that light detectors detect individual photons at specific times. Either a photon is detected or it is not detected, independent of any other photons emitted by other atoms. Similarly, an electron detector either detects one particular electron at one place at one time, or it fails to detect it. There is no such thing as electrons doing constructive or destructive interference, they are always detected entirely independent of each other. ;)
Light is not waves. If you want to find out what it is, study Quantum ElectroDynamics, which will teach you how photons travel in ways that exactly 100% mimic light waves, without being light waves. You can't understand it unless you study quantum electrodynamics.  And since light isn't waves, there's no reason to think of electrons as waves either. They sometimes exhibit some wave-like properties, that's all. Diffract them, bounce them off mirrors, things like that. People who can't accept all this will never be successful physicists. ;)
