Does bench pressing work out the same muscles as push-ups?
If not, What's the difference?
I'm sorry, but the selected answer for this question is terribly ill-informed. It doesn't even answer the OP's question of which muscles are actually utilized for each of these activities.
Common myth #1: The pushup is merely an exercise for muscular endurance.
Common myth #2: The bench press is far superior to the pushup for building muscle, mainly due to the ability to quickly and easily adjust resistance.
The problem with the bench press is the bench itself. Your upper body is rendered immobile by the bench, preventing your core from doing any work to stabilize your body; it's all done for you, which is a bad thing. Many professional bodybuilders agree that if you don't have core strength, you have no strength at all. The pushup demands core strength, while the bench press ignores it.
The bench press also ignores the serratus anterior (muscle group on sides of chest from 1st to 8th ribs,) while the pushup, again, demands it.
In fact, doing a proper pushup engages every muscle between your shoulders and toes. Including abdominal, hip, thighs, etc. The bench press can certainly chisel your chest, but little else. You are conserving energy by using far less muscle groups, but you are chiseling far less of your body. So you will end up looking disproportioned if the bench is all you care about.
There are also risks involved with the bench press that are moot with pushups. The bench press will put increased strain on ligaments and tendons (not good,) and can even damage the cartilage pads between your joints (very bad.)
Let's also stomp the misconception in this thread about not being able to apply additional resistance to a pushup. This can easily be accomplished through stacking plates on one's back, wearing a weighted vest, wrapping chains around the torso, elevating one's feet, using one arm, well you get the idea.
Quick anecdote: I know a guy who was doing 350+ on the press, but tore a rotator cuff whilst throwing a tennis ball to his dog. Why? Because all he cared about was the bench. He believed he was far stronger than he actually was simply because of the size of his arms and chest.