Ok, feel free to correct me if I'm wrong, but isn't the thing that makes the Higgs so important simply that it's the only particle predicted by the SM that hadn't yet been found?
Sort of. Ultimately, it's not actually about a specific particle at all, but about symmetries. The standard model in it's basic form has certain symmetries which result in predictions that don't actually agree with observations. In order to get the theory to match things like the mass of certain particles, you need to add some way that the symmetry is broken. The Higgs mechanism adds an extra field (the theory is actually made up of fields, see
here) into the SM which allows breaking of the symmetry in the correct way to get the masses for the W and Z gauge bosons that we actually observe.
So it's not just a question of there being a missing particle, this is essentially a test of the fundamental basis of the whole theory. Without the Higgs mechanism the entire standard model just makes no sense at all, so if we don't see the extra particle it also predicts the whole thing pretty much falls apart.
And this discovery basically elevates SM to the same status as GR--a theory that's withstood every test we could throw at it?
Sadly not. Even with the Higgs, the standard model is known to be seriously flawed. The standard model doesn't even attempt to cover gravity, it can't explain dark matter, it fails badly at explaining dark energy, it fails at explaining the matter/antimatter imbalance, and so on. In fact, it's not even proven to be mathematically consistent. Relativity has some trouble at certain extreme scales, but the rest of the time it seems to be pretty much a perfect match to reality. The standard model just plain doesn't work. It's the best we can do so far, but we know it's not even close to complete.
I mean, it's nice to know that it has something to do with some form of mass, but that's not really why this is such a big discovery, is it? It's not the HB itself; it's the fact that it's the last piece of the puzzle. (At least within the domain the SM covers.)
Discovering new fundamental particles is pretty big news that tends to result in Nobel prizes. The W and Z resulted in Nobel prizes, neutrinos resulted in Nobel prizes, the tao resulted in Nobel prizes, quarks resulted in Nobel prizes. In fact, the only ones that didn't are photons and electrons, since we already knew about them, and muons, where one of the discoverers had only just got a Nobel prize that year anyway. So yeah, discovering a new fundamental particle really is pretty important stuff just on its own.