you do use a lot of stabilization in a push up, thought not in the same way as with the bench press. In the bench press you're stabilizing the bar, in the pushup you're stabilizing the body. That's a bit of a gross generalization, but it's pretty much it. The bar is the main unstable thing in the bench press, since you're lying on your back. In the push up, though, your body is the unstable thing that needs to be stabilized.
On a final note, though, I don't really think we should compare the push up and bench press that much. They are different exercises.