When you vote, you get either a "pick your favorite from the list" choice or an "express your approval of this item" option.
Vote for one:
 Frederick J. Flontstein
 Barnabas P. Rubbhauser
 Dean O. Sauris
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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.