The difference is that a helmet visor is against the rules, but this has already been discussed.

Seriously though, I used to wear a helmet visor, and I found it to be inadequate. A cycling cap has the benefit of soaking up sweat and preventing it from dripping in your face. The visor can also be rotated/adjusted and the brim flipped up and down depending on the specific angle of the sun, vs a helmet visor which is always fixed in one position. Also, they are suitable to wear at the cafe/bar after you stop for a break, so you don't have to sport your messy, sweaty hair in public. And they look cool.