The phrase "a bit" is an adverb, used to modify an adjective or an adverb. The prepositional phrase "a bit of" is used as a quantifying adjective, and must modify a following noun (just as all prepositional phrases take a noun as an object).

Both mean "a small amount". "A bit" is usually used at the end of a sentence in response to an answer whereas "a bit of" is followed by an noun. Question: "Do you want some pizza?" Answer: "Just a little bit" Statement: "I gained a little bit of weight this winter!"

*"response to an question"