For prime lens, it can have a bigger aperture (f1.2, f1.4, f1.7, f1.8) while for zoom lenses, the maximum it can go is f2.8.
Due to this, the DOF is affected hence people tend to prefer the prime lens, besides the 50mm lens is one of the cheapest fast lens one can lay their hands on.
Prime is cheaper, lighter and usually sharper though I can say zoom is matching it very closely when N14-24 came out.
It is difficult to control the quality of a zoom, thus for a zoom. Example, you might get a sharper/ distorted picture at 17mm but not 25mm but it changes again at 50mm 55mm.
Actually with the barrage of revised and really sharp zoom offerings by various manufacturers, the old debate of better IQ of primes vs zooms have really narrowed to a close. I for one cannot tell the difference if one is taken by a zoom or prime. For me the idea of choosing one over the other doesn reside much on IQ arguements ( but thats for me ).
DonnyDan has cited the classic example of Nikon 14-24mm zoom that puts the 14mm prime lenses to shame.
My main consideration is actually prime lenses often offer ya a larger aperture compared with zoom lenses.
Not so obvious differences: You'll notice that prime lenses often have a much bigger aperture like f/1.8, f/1.4 not found on zooms. This is very useful for subject isolation (bokeh / blurring of background) which is often what many portrait photographers desire.