When you vote, you get either a "pick your favorite from the list" choice or an "express your approval of this item" option.
pick-your-favorite voting:
Vote for one:
[] Frederick J. Flontstein
[] Barnabas P. Rubbhauser
[] Dean O. Sauris
express-approval voting:
vote for this answer by clicking the up-arrow next to it!
When you rate, you get to assign the thing in question to some particular point along a scale:
1(bad) 2 3 4 5 6 7(great)
please rate the service today: o o o o o o o
When you rank, you place several objects in order of preference relative to each other:
My personal ranking of the four major citrus fruit flavors would be orange, lime, lemon, grapefruit.
For your application, I would say vote is the best choice.