also note that prime lenses will normally be much sharper than zoom lenses (in the same quality and price range).
Plus everything Christie said
but they lose out on the versatility that a zoom has - and if you buy a high quality zoom (like a Canon 70-200mm) then the results will be more than sharp for use.
I feel like primes also unlock some creative thinking patterns that get locked up by zoom lenses. Like angles, and moving around more and all that stuff.
The $80 Canon 50mm f/1.8 beat every Canon L zoom (all over $1000) it was tried against in resolving fine details in a Pop Photo (I think that was the mag) lens shootout a few years back.
It's easier to make great fixed focal length lenses; the design is fairly simple, and the basic concepts were mastered 100 years ago. Really good zooms were pretty rare even 30 years ago. It took computer aided design to work out the kinks (not all of which have been worked out).
Each has it's advantages; I wouldn't want to give up my primes or zooms.