I seem to recall a particular line in Fellowship of the Ring (and, after some Googling, perhaps in some of the other movies) which may give us a good reason:
ARAGORN: Legolas! What do your elf-eyes see?
This recurs in several other places, such as in the first half of The Two Towers, where Legolas and Gimli are in Fangorn forest:
'Yet here we are-and nicely caught in the net,' said Legolas. 'Look!'
'Look at what?' said Gimli.
'There in the trees.'
'Where? I have not elf-eyes.'
And a bit later (emphasis mine)
It seemed to Legolas, as he strained his farseeing eyes, that he caught a glint of white
This gives a strong implication that Elves have super-human visual acuity, which would naturally make them better at using any sort of ranged weapon - bows included.
