Can human ear distinguish between sounds of different phase? A noise cancelling headphone produces a sound that's out of phase with ambient noise, mixes them and hence cancels them.
But I wonder if we were to hear the 2 sounds which are out of phase to each other, will we be able to feel any difference.
If we can distinguish, how does an out of phase noise of finger snap sound like?
 A: For one ear alone, the sound you hear will be the sum of all sounds at that point. So there's not really such a concept as "out of phase" there... "out of phase" relative to what? If you have two speakers generating two sounds that are identical except for a 180 degree phase offset when they arrive at one ear, the perceived volume at that ear will be zero or close to it.
The story becomes a bit different when you talk about two ears. If your ears hear the same sound but it has a slight phase offset in one ear compared to the other, your brain uses this (plus a few other factors) to judge the direction to the source of the sound. E.g. a sound coming from a single point off to your side is likely to have a slightly different phase at each ear, this is part of the set of info your brain uses to figure out where the sound came from (as well as e.g. frequency filtering from the shape of your ears, amplitude differences, visual information, logical conclusions about what "makes sense" in the current situation, etc.)
So, if you're asking if one ear can distinguish between "sounds of different phase" from a single source that doesn't really make any sense.†
If you're asking if one ear can distinguish between "sounds of different phase" from multiple sources, not really, you're only really aware of the end result.
If you're asking if you can distinguish between "sounds of different phase" across both your ears, yes, you do it all the time, it's one of the things that helps you locate the source of a sound.
For the one ear, case, though, it's a lot easier to identify two sounds whose phase is changing relative to eachother, it's a common, distinctly recognizable audio effect especially with guitars, you probably recognize the sound, e.g. https://youtu.be/pvScdOldfc8?t=154.

† By "doesn't really make any sense", I actually mean: You wouldn't be able to tell unless you knew what the sound was supposed to sound like "normally" as a reference for comparison. There'd be nothing inherently identifiable about such a sound, you'd need a mental reference. If I played two identical waveforms, overlapped but with one shifted slightly, of a sound that was completely unfamiliar to you, you would not be able to identify that as any kind of "phase shifting" - it's just a waveform like any other, but if I did it to a human voice, you'd be able to tell something is odd, because you know what a voice should sound like.
A: 
But I wonder if we were to here the 2 sounds which are out of phase to each other, will we be able to feel any difference.

This sounds like a hypothetical situation. Step in front of two speakers that play the same signal. Due to the different lengths from your ears to the speakers, you hear two out of phase versions of the signals. If you move your head or entire body, the phase difference changes. I can't hear that.
For the most part, the brain calculates the direction where a single sound is coming from from the phase difference of what your two ears hear due to their different locations on your head.
If you have two signals, you get a phase difference for each one. This is used in signal processing for stereo enhancement: add the left signal with reduced volume and phase shifted (delayed) to the right signal and vice versa. This way your brain is tricked into detecting the directions of the two stereo signal channels. Thus you feel the left signal coming from the left. 
tl,dr;
Phase shifted signals are interpreted as directional sound. Two such signals can be interpreted as two such sounds.
A: I have a Hi-Fi test disk which illustrates one situation in which phase has an audible effect.  It has a test which can determine whether your speakers are connected correctly.  First it plays white noise through both speakers in phase.  If your speakers are connected correctly and positioned sensibly then this sounds as if it is coming from a single source between the speakers.  Next, it plays the same white noise but with the speakers now out of phase.  It sounds as if the noise is coming from the two speakers and not one middle source.  If one of your speakers has its connections reversed then the results are reversed.  So your ear, well mine anyway, is fooled into interpreting the in-phase signals as a single middle source; this is idea behind stereo.  It is not fooled by the out of phase signals and recognises them as separate sources.
A: I'm not sure I'm quite addressing the question, but my crude understanding is that a single ear behaves much like a sonograph and responds to the amplitude at various frequencies with little regard to their relative phases. The window over which this occurs is a few tens of milliseconds. Phase-shifts corresponding to time delays larger than this become very apparent. 
The degree to which phase distortion can be perceived is a topic of some argument with audiophiles claiming to hear differences that most people would never notice. There are various mechanisms in the ear that result in nonlinear distortion and the intermodulation of frequency components particularly at high amplitudes. For example, a beat frequency can somettimes be perceived if a harmonic is slowly continuously shifted in phase with respect to the fundamental. 
This article has some good discussion and references.
A: A click of fingers or any snap sounds probably won't work because they are not pure tones, they are a mix of frequencies.
From The Sound Of Silence

If you use a signal generator to send a steady sine wave signal to your left speaker and simultaneously send the same signal to the right speaker but with it's polarity reversed when you place your ear at the exact point between the two speakers you should notice that the volume has dropped down to an inaudible level. (You can't have both ears at the exact same point so I mention just one ear).
In free space this should work for any sound signal but in practice you're likely to be operating indoors so reflections from the room surfaces will also superpose at the listening position. Whether those signals would interfere destructively or constructively depends on their wavelength and the distance the sound has to travel. Because it varies with wavelength the reflections of a complex sound such as the human voice won't have a simple cancellation effect but will instead produce something similar to a comb-filter with some frequencies cancelling and others not (at a given point in space). This is why it's easier to demonstrate the cancellation with a simple sine wave. It only has a single wavelength so you have clear nulls that you can find by walking around the room. 

A: "Out of phase" sound can actually be a lot of different sounds, varying from a little bit out of phase to complete counter phase. Counter phase sound being more or less the acoustic mirror image of the original sound.
Separately you can't hear the difference between them, but your microphone can. Long before there were sound cancelling head sets, phase shifting was used to keep the sound from monitor speakers on a stage interfere with the microphones. Back in the day when people could still sing, hearing yourself as a singer was what kept you from missing the tone. Just having the original sound blow in your face would cause it to 'sing' around at a very loud and high screechy pitch. Having the sound of the monitor out of phase would solve that problem.
Some performers used that effect to give a little something extra to their music, primarily by keeping their guitar smack in front of the monitor. Bands like "the Who" and "Ten Years After" were well known for that. Pete Townsend would actually smash his guitar to bits on the monitor, which didn't do the sound any good in my opinion.
So no, you don't hear or feel the difference, but you can play with it. Synthesizer technology does so in order to create new sounds and new sound effects. The snapping of a finger is as good a sound as any to work with, as far as that's concerned.
A: The sensation of the sound comes from our ability to distinguish the source of the sound based on the variation in the perceived volume of the sound between both the ears. See this video for a visual representation.
