The human ear can easily distinguish quarter tones. Some 20th century composers have composed in microtonal music (Partch uses a scale of 43 tones as opposed to the usual Western 12). In Western tradition the modern temperament wasn't always used and there could be a significant difference between G# and Ab for instance in earlier times. But in modern Western music we have standardised semitones and you automatically translate slightly sharp or flat notes to the notes you're expecting. (String players and singers tend not to realise notes at their exact equal temperament but do adjust them. I seem to remember some research that shows that this is perceived as sounding better than exact equal temperament realisation).

If you want to try and hear quarter tones to see if you can do it, get a guitarist to bend a note on a string for you. They can easily bend it a quarter tone sharp and you'll hear the difference between that and the note at the next fret. (A whistle player can do something similar for you bending the note).

The current Western scale is not the only one is use and you find all kinds of other scales around the world with intervals that don't correspond to the tones and semitones of current use. And as Mary says you only have to listen to some of the traditional recordings to hear how quarter tones have (and still are) in use in English traditional music.

Mick