Feathers
Both bats and birds fly by rapidly moving their wings through the air to create lift. They push against the air. To do this, their wings need a relatively large surface area, but not a lot of weight. Bats accomplish this with extremely thin skin in their wings that is stretched out over a large area and able to generate a significant amount of lift without adding much weight. Birds have feathers, which accomplish the same thing.
Fingers
Neither birds nor bats have fingers in the sense that we traditionally think of them, but if you look at their skeletons, you can see that they each possess the equivalent of fingers in the structures of their wings. Bats, like us, have a thumb and four fingers. The thin membranes of skin in their wings are stretched out on this framework of five digits. Birds have a thumb and just two fingers, which are much shorter than those of bats.
Wing Strokes
While both bats and birds are working with the same laws of physics and aerodynamics to achieve flight, there are differences in the specific mechanics of how they go about it. Scientists have studied these differences using fog and lasers to analyze the air currents surrounding the flying animals. As birds swing their wings upward, the feathers separate, preventing lift-reducing currents of air. Bats, which use stretched skin and not feathers, cannot do this. One practical result of this is that birds can generally fly faster than bats, but bats have greater maneuverability than most birds. With some notable exceptions, like the hummingbird, most birds cannot hover like bats can.
Independent Wing Motion
Bats are able to steer in flight by independently using one wing or the other, taking an extra stroke with the right wing to veer to the left, for example. Birds do not do this. Their wings are synchronized and they make use of their tails for steering. Analysis of air currents show that bats in flight create two separate vortices, one behind each wing, while birds in flight produce only one vortex for the whole bird.