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Can use a lot more weight, so it's ideal for the low rep anaerobic range. It's for mass building.
Can be assisted by a spotter, so you can go to failure. This is good for breaking plateaus and stimulating growth.
More dangerous if you don't have a spotter. The bar can fall on your chest if you have no more strength to get it back up.
You can do incline, decline, or flat bench press which targets upper, lower, and entire chest respectively. A push up is the same angle as a flat bench press.
Only uses 50-70% of the body weight depending on your body proportions, so you're going to do higher rep aerobic range. It's for cardiovascular conditioning.
Wrists have to do a little bit more work because of the bent position.
Allows for hand position variations than bench press. For example, placing hands near the abs turn the pushup into a tricep workout. Another example is the diamond pushup, which puts more emphasis on inner chest and tricep.