Now the thing you have to understand is that punches should travel in a straight line [with the exception of hooks], when it does and your wrist is not in the same line, it could bend at impact and break. The thing that is good with punches is that they generate power all the way into the center of the target. I recommend that you hit with your knuckles squarely on your target.
Now, elbows needs torque. Elbows lack the penetrating power of the fists but is more powerful and is a lot sharper than fists. Don't hit squarely with an elbow and you can draw blood from your opponent.
The reason why many believe elbow hits harder than a punch; It doesn't hurt as much doing an elbow when compared to punch. Try punching a wall, it hurts. Now do the elbow, it doesn't hurt as much. It's actually quite all about physics. The longer the tip of the object, the more powerful will the impact be. Now why does elbow hit so hard? The turning motion of the body, shoulder, elbow and hips. That turning movement is a lot more powerful when doing the elbow then when doing a punch. That's the reason why elbows hit hard. The human body is a complex piece of machinery and sometimes, when things looks obvious, it isn't. It doesn't hurt as much when hitting hard with an elbow, so many can concentrate a lot of force behind it and smash away while with your fists, it will hurt and thus you won't put much power into it.
Elbow and fists can hit just as hard, it just requires training.
Now, why would you use elbows over fists? Range. Already explained.