At their core, both terms are very similar. There is a slight difference though, which is also touched upon in this reddit comment.
In an aggro deck, every card is supposed to come out quickly and put pressure on the opponent. The term is very well explained in the question you linked, so I won't get too specific, but what I want to emphasize is that the aggressive nature of the deck means every card supports the aggressive mindset and tries to do something immediately.
Rush decks, on the other hand, are defined by their goal of winning quickly. While this usually means being aggressive (in fact, most rush decks also qualify as aggro decks and vice-versa), it can sometimes mean trying to get to a cheap combo quickly. The important point here is to win as quickly as possible, giving the opponent a minimum of time to setup some sort of big strategy.
Current Hunter decks are an example, while they try to win quickly and put the opponent on a clock, they don't necessarily do so aggressively by playing lots of minions and attacking, but can sometimes wait a few turns just doing damage with their hero power and Eaglehorn Bow, using Secrets to keep the Bow on their field, waiting for everything to fall into place for their big Unleash the Hounds combo.
Basically, "rush" refers to a kind of time frame (early/mid/late game), the others being midrange and control, while "aggro" refers to a strategy (e.g. rush, combo, aoe, big guys, ...).
So, to summarize:
- have an aggressive playstyle
- use every card as quickly as possible
- put pressure on the opponent at all times
- can choose to trade favorably instead of doing as much damage as possible
- try to win quickly
- sometimes hold on to cards and wait for combos
- put the opponent on a clock
- nearly always do as much damage as possible