Are these the same things? Almost. A drop shot typically is made as short as possible, and is usually used against a chopper caught off the table. The goal here is to keep the ball as short as possible so that the defender either can't get to it (an "ace"!), or is lunging to get it, and either can't make a good return or recover for the next shot. It should also be low, both so the defender can't run in and hit it in, and because a higher ball stays in the air longer and so gives the opponent more time to react to it. (You can also drop shot against an off-table lobber with a dead block or chop block, though that's more difficult to keep short.)
A short push against an attacker's serve is actually more effective if it isn't too short, but short enough so that it would bounce twice if given the chance. The problem here is that the attacker is usually at the table, and so dropping the ball very short won't catch him out of position unless he's very short or immobile. And if he's in position for the shot, the shorter ball is both easier to attack with a flip than the slightly longer one, and easier to drop short.
There are always exceptions. Against a very fast-footed defender that you can't really catch with a drop shot, you might not want to drop the ball too short as it just gives him easy balls to pick hit, especially with a running backhand smash. And against an attacker who likes to serve and step back a bit, looking for a long return, an extra-short drop shot might be effective. But as a general rule, use drop shots against choppers, and short pushes against attackers.