Is there any difference in the plot/way of the gameplay playing with another character? Or ir just for the sake of playing with more than one character?

No story difference as I recall. Concerning the gameplay, I think the only difference that I recall is that the combat is a bit different (regarding the kick). I didn't like how Emily Hartwood fought.

A small easter egg, regarding a photo is the only difference.
If you use search near that photo, Emily says it's a photo of her and her late uncle. Carnby says something different, but I didn't bother playing as him.