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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.
Push up
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.
Don't forget about incline and decline pushups to target different muscles. Really the only main difference between pushups and bench press is the amount of weight. – Chris PietschmannMay 15 '11 at 16:09
Both can be used aerobically or anaerobically depending on variables such as load, sets and rest intervals. One nice thing is that you can do push ups anywhere! – Christopher IckesMay 15 '11 at 19:21
Since I usually work out alone, I do my bench presses using a squat rack. I set the pins so that the weight stops just above my chest and I can safely do the exercise to failure without a spotter. Just make sure that you set the pegs high enough that the weight doesn't rest on your chest at all. Depending on the adjustability of the rack and the bench that you're using, it might not be possible to find a good height, so this can be hit or miss. I've been lucky at most of my gyms in the past. – AardvarkMay 16 '11 at 18:54
The squat rack doesn't work the stablizer muscles – JoJoMay 16 '11 at 19:38
Bench pressing with dumbbells is another option if you don't have a spotter as at failure you can drop them to your sides. – AlastairMay 19 '11 at 10:23