What function do the butterfly and bird wings differ?

The functions are exactly the same, to let the creature fly. The way a bird flies is a bit different than the way a butterfly does. A bird, essentially swims through the air, much like you doing a breast stroke in the water. The wings going forward have a low resistance to the air, the down beat, has high resistance because of the feathers. Kind of like extending your arms, and making small circles, now imagine if you had feathers, pushing against the air on the down stoke.

A butterfly, seems to "slither" through the air. Much like a snake moving through the grass, the wings form a wave pattern. Like the bird, extending the flat thin part, but like a snake, creating a wave like motion. *get a blanket, and you and a friend each hold an end, and alternate moving your ends up and down, to make a wave. Note the wind created as the blanket moves. This, incidentally, is also the method a manta ray and Spanish dancer use to swim in water

Since, apparently links aren't allowed, I will add that there is a video on YouTube, of a butterfly taking flight in slow motion, that nicely shows the wing movement.