Why is bench pressing your bodyweight harder than doing a pushup? Why does bench pressing your own bodyweight feel so much harder than doing a push-up? 
I have my own theories about the weight being distributed over multiple points (like in a push-up) but would just like to get a definite answer.
 A: It's harder because you are benching more weight. Benching your bodyweight requires your upper body to complete 100% of the lift. 
When your feet are on the ground and you are pushing yourself up you are only lifting approximately 1/3 of your body weight. You should be able to do much more push-ups than repetitions of your weight on the bench press. 
Consider this, is it easier to get off the couch or squat your body weight? The answer has the same basic principle, the more an object weighs the harder it is to lift.
A: Consider leverage.  Assume as in @Michael's comment that the centre of mass is somewhere near the middle.  Further assume that the toes are fixed to the floor. If the torso and legs are rigid the centre of mass does not lift as far as the shoulders (at which the pressing force is exerted), so you've got leverage in your favour. This probably reduces the lift to about 2/3 or 3/4 of what it would be if you didn't put any weight on your feet.
A: When you're doing press-ups, your arms have a mechanical advantage over the mass of your body. The force that they exert does not pass through your COM and, instead, exerts a torque on your mass with your feet being the fulcrum of the lever.
When you benchpress, your arms are exerting a force that passes straight through the mass of the weights. As such, there is no torque and no mechanical advantage.
Put another way, with each exercise, your arms move the same distance x. With a benchpress, the weights move this full distance, but with pushups, your COM does not move the full distance; thus, the work done during pushups is lower.
A: When doing pushups, you're not lifting all of your body weight the full distance. Your heels don't move any appreciable amount and a point halfway between shoulders and toes only moves about half the height.
Consider an iron bar, 1m in length. Lifting the whole thing 1m requires twice as much energy as lifting it half a meter. Lifting one end to a height of 1m and leaving the other end on the floor means lifting the mass half a meter on the average.
A: When doing push-ups, you are making your body into a lever! Your feet are the fulcrum. So you get the mechanical advantage that makes levers useful. It's just like how lifting the handles of a wheelbarrow (pivoting on the wheel) is a lot easier than simply picking up the contents of the wheelbarrow.
For similar reasons, doing dips with your legs freely dangling is a lot harder than doing a bench dip where your legs are extended with heels on the ground.
A: While doing push-ups, you don't push your whole body weight. You have your toes on the ground, so your body weight is distributed between your feet and your arms.
While benching, you have no support from feet. You hold the whole weight with your arms, so benching your body weight is always tougher. 
