Here's a modern viewpoint: there are many different things we can mean in saying that one model of computation (e.g., quantum) has abilities that exceed another model's (e.g., the polynomial hierarchy). So, we
could be talking about the number of computational steps, the first thing that complexity theory tried to understand and also (by a twist of fate) nearly the hardest. But we could also be talking about several other things, like communication complexity for a distributed problem, or
query complexity: the number of accesses that need to be made to a very long input, which is more fancifully called an "oracle." For each of these settings, we could be talking about decision problems, promise problems, relation problems, or exact or approximate sampling problems.
Think of all these separation problems as terrifying monsters that we have to defeat. Some monsters, like P vs. NP with no oracle, are clearly the bosses of the entire game. But in the meantime, we just try to slay whatever monsters we can, because that's how we build up our stamina points and learn ins and outs of the game. And also, almost all these monsters live in dungeons that you can only reach by slaying easier monsters first.
For now, unrelativized separations, like P!=PSPACE, are monsters that we almost never have the tools to defeat. They're like 30 levels ahead of us in the game---or maybe 300 levels; one thing about this game is that it never tells you how many levels you still need to clear.
Oracle separations---and communication complexity separations for that matter---are easier monsters. But the point I want to make is that they're clearly, unequivocally part of the same game as the harder monsters that we're trying to get to.
This is particularly clear in the case of oracle separations from PH. When theoretical computer scientists proved in the 1980s that there's an oracle relative to which PH!=PSPACE, they did so only by proving the inability of small constant-depth circuits of AND, OR, and NOT gates, so-called AC0 circuits, to compute the n-bit PARITY function. This is universally considered one of the most important lower bounds on circuit size ever shown. (Circuit size lower bounds, of course, are the dungeon that P vs. NP is the boss of.)
The connection is this: a PH algorithm that queries an exponentially-long oracle string can be seen as just a massively scaled-up AC0 circuit reading its n-bit input (with no oracle). Conversely, an AC0 circuit is just a massively scaled-down version of relativized PH. There's a giant monster that you have to fight with giant swords, and a tiny monster that you have to fight with tiny swords, but the two battles are isomorphic. Win one and you win the other.
In the same way, proving an oracle separation between BQP and PH, is exactly the same challenge as giving an ordinary computational problem (with no oracle) that quantum computers can solve in polylogarithmic time, by querying the n-bit input in superposition, but that's not in AC0.
Of course, eventually we'd like to prove that interesting problems like 3SAT have no polynomial-size circuits of any kind, thereby establishing P!=NP. For now, though, a little beyond AC0 is the frontier: the most complicated place where we still know how to prove lower bounds on circuit size. And for problems that have the sort of structure that comes with being solvable in quantum logarithmic time (basically, being approximable by low-degree real polynomials), we didn't know until a month ago even how to put them outside AC0, let alone the classes slightly beyond AC0. So an important monster has been slaughtered in the frontier region, and we can now advance to a previously unexplored corridor, in the huge sprawling dungeon where somewhere P!=NP is lurking.