Like tekromancr was getting at, you get a logical contradiction if you assume the existence of 1/0, if you also assume the ring axioms. To elaborate, if 0x = 1, then 0 = 1. It is also an axiom that 0 != 1, and even if you abandon this axiom you only get one extra ring that only has one element. Boring!
You also get a logical contradiction if you add a square root of -1, if you are assuming, say, the ordered ring axioms. But you can lose the order structure and then there is no more contradiction.
If you want to add a new number x with 0x = 1, then you also need to propose how one might alter the notion of a ring so that you don't get a logical contradiction, and you are still left with something interesting. Just like how we altered the notion of an ordered ring, by removing the ordering, so that you don't get a contradiction when adding a square root of -1.
So the crucial difference is, as far as I know, nobody has succeeded in doing this when it comes to division by zero.