Here’s a quick video discussing the difference in movement between the push-up and bench press. In a push-up, you want to end the movement by pressing as far away from the floor as possible. This helps engage some key shoulder stabalizers like the serratus anterior. Doing push-ups in the fashion, the proper way, will help keep your shoulders healthy in the long run and are a great assistance movement to the bench press.
The bench press though, does not use the same mechanics. While bench pressing, your shoulder blades should be locked back and your lats should be as tight as possible. During the downward portion of the lift, you should think about actively pulling the bar down to your chest. This will help engage those muscles. Once you begin to press the weight back up, do not lose the tension you’ve just developed in the back of your body. This will not only help you move more weight by providing a stable base to press off of, but it will keep unnecessary stress of the front of the shoulder by holding it in a better position. This technique is true of any bench press variation as well including dumbbell bench press, floor press, and the incline press.
Check out the video below for further explanation: