People keep saying this but I think it's a bogus distinction. They're both introductions to their respective games and aimed at new players, aren't they?
Yes they are.
But it is fair to say that they go about it in
very different ways.
The Pathfinder box is, essentially, a complete (albeit
very limited) RPG. For that reason it has very little pre-gen adventure material, but it does include character generation (in addition to pre-gens), monsters not in the adventures, and some guidance for the DM as to what to do next. A group could, in principle, buy that and then play indefinitely with nothing else - though I suspect they'd get bored before too long.
The D&D box, by contrast, is essentially a fairly big adventure with some surrounding materials. It's approach is basically to provide the DM with "My First Adventure", and then guide him on to Basic and/or the "big three". (That shouldn't be a controversial statement - Mearls basically
said as much.)
So, which is better?
Well, I'm in the happy position of having both and liking both, and my opinion is: it depends. It depends on who you are and who the gamers around you are.
If you're an existing gamer, the D&D set is better, but...
you should get it if and only if you intend to run that adventure (or, I suppose, if you
really need monsters, and can't wait for the update to Basic later this month). The thing is, Basic includes everything in the rulebook, the pregens can be downloaded in PDF form, as can the blank character sheet. So the only thing in the box other than the adventure is a set of dice.
If you're a new gamer (who, somehow, can't just join an existing group or find someone to teach you), you should look around at the groups near you. If they're Pathfinder guys, go for that; if they're D&D guys, go for that.
And if you're a new gamer and there are
no groups around you (and yet you're here; not sure how that works

)... I don't know. My gut feeling is that the Pathfinder set makes for a better first step*, but that
D&D (rather than the Starter Set) has a much easier jump to the 'real' game.
* I do need to note that there may be a bias at play here - the Pathfinder set much more closely resembles the Red Box I started with, so I know it can work.
About the price: Yes, the Pathfinder set is considerably more expensive. It also includes many more reusable components - a strudy battlemat, character pawns, etc. Indeed, these may be sufficient justification to buy the set even if you never intend to use it as a starter.
But, actually, I would argue that the price isn't that important a factor, because the value of either box is so dependent on other factors - if you're aiming to have a nephew join your D&D game, there's no point in getting him the Pathfinder set (or vice versa). Or if you're not intending to use the specific adventure in the D&D set, there's no point in getting that set, regardless of price. And, finally, some people will just learn better with a how-to guide, and some will just learn better with a single extended example - the former would do better with the Pathfinder set; the latter with the D&D set.
(Personally,
I'm much more likely to use the D&D Starter Set; if I were buying for my nephew, though, I'd get him the Pathfinder set. And, yes, I know that's not terribly helpful.)
Finally: A few years ago, we had gone more than a decade without a good starter set at all - there had been plenty produced, but they'd been pretty awful. We now have two (or four, if you include Star Wars and Dragon Age). I like to think that's a really good thing.