Push-ups and bench presses are both upper body strength exercises which target the chest, shoulders and triceps.
But they also differ in several ways:
Body Position: In push-ups you use your own body weight as resistance and you're in a prone position with your hands on the ground and your body parallel to the floor. In bench presses you lie on your back on a bench and lift a barbell or dumbbells.
Equipment: Push-ups require no equipment other than your body making them one of the most convenient strength exercises you can do. Bench presses on the other hand require a bench and weights (barbell or dumbbells) to perform.
Muscle Engagement: Both exercises work the chest, shoulders, and triceps and are good for building muscle and increasing strength. However, unless you are using a weighted vest, resistance is limited by your own bodyweight during a push up while progressively heavier weights can be added to a bench press.
Core Engagement: Push ups double up as a good core training exercise as you must maintain a strong posture throughout the exercise. There is much less core engagement during a bench press as your weight and posture is largely supported by the bench.
Difficulty: Biomechanically both exercises are fairly simple to learn. The advantage of bench presses is that really light weights can be selected so you can master good technique without the pressure of heavy loads. The difficulty of push ups is determined by your own bodyweight. If you are carrying a lot of extra weight then push ups might be very (or too!) demanding. On the flipside if you are very light then push ups might not provide much of a strength challenge without the addition of some external resistance e.g. weighted vest or band.
Variations: Push-ups offer various modifications and variations e.g., incline, decline, one-arm push ups, while bench presses have variations like incline, decline, and close-grip bench presses.
Safety: When performed with a bar and without a spotter the bench press is one of the more dangerous exercises you can perform. If training alone you can become pinned under the bar and/or there is potential for the bar to fall on your face or neck. Push ups on the other hand are a much safer alternative with a very low degree of risk.
In summary, push-ups are a great bodyweight exercise which can be done anytime, anywhere while bench presses involve lifting weights on a bench in a gym or home setting. They both target the same muscle groups and can be part of a well-rounded strength training routine.