When used alone, "little", "bit", and "little bit" all mean "small". But if you emphasize "little bit" over "little" or "bit" then you are emphasizing HOW SMALL it is.
Example1: "John is a bit sick." Here you're just saying that John is sick, but no so sick.
Example 2: "John is a little sick." The same here.
Example 3: "John is a little bit sick." Here you are emphasizing how small the sickness is. In this case John isn't as sick as in the first two examples.
"The movie was a bit different this time." and "The movie was only a little bit different this time." In the both cases the movie was different, but in the second one it was way less different than in the first one.
I hope it makes sense.