There are different opamps, yes, and there are different circuits. All the [solid state] circuits that aren't designed by Kevin Gilmore—and are also discussed on this forum—use opamps (though not all allow for opamp rolling). Some circuits are better than others, yes. Basically, the way it works is, no circuit element is perfect. If they were, then you wouldn't need an amp. A pot would do just fine. (Actually... some audiophiles just use a pot for their "preamp", and they call it a "passive preamp" even though it isn't a preamp at all...). The better circuits are ones that account for non-idealities better. Hmm... I'm not the best person to explain this stuff, but since this isn't in the DIY forum, here goes...
So you start with the cmoy. It's got an input buffer in the form of a cap, and a totally unbuffered output. But it sounds ok for the price: it's hella better than the output of most portable electronics.
Next step up is the mint (for simplicity, I'm going to stick what's on Tangent's website; also, I'm going to totally ignore the powersupply sections, which is pretty stupid, I admit, but I don't understand that stuff nearly as well). It uses—like the cmoy—a dual-channel opamp and has a cap on the input, but there are a couple big improvements. First, you can buffer the opamp into Class A with a CRD, and second, there's a BUF634 current buffer on the output. Tangent has a decent description of what Class A biasing is, and why you want it, so I suggest you read that if you want to know more. The latter enhancement fixes the fact that opamps are crappy current sources. Yes, grasshpr, they suck as current sources, which is why every amp over $100 has some sort of current buffer between the opamp and the output.
The pimeta can use a JFET cascode to bias the opamps into Class A and has a slightly more interesting input stage. But the most important thing (aside from the powersupply, which looks somewhat more beefy) is that it has an active ground. It's a more symmetric design in that, usually the ground just sits there, but in the pimeta (and the PPA), it can push current back through whatever it's driving. That's fine for passive loads (like headphones), but apparently it oughtn't to be used as a preamp.
The PPA seems to benefit from a much, much beefier powersupply, and optional "aftermarket" buffers (oh, and it uses single-channel opamps, unlike the previous amps). Standard on the PPAv1 are Intersil buffers (up to four of them), but most people seem to agree that the biggest upgrade one can get for it is a discrete output stage (that uses transistors!).
The M³ is based on the PPA, but makes some significant changes, the most obvious of which (to me, anyway...) is the use of MOSFETs on the output.
Opamps can change the sound, but they're victim to any limitations of the circuit you're using. Opamps can also put constraints on the design (some aren't stable at low gain; others need lots of voltage or they'll clip). Opamps vary in price, yes, but the big part of the price is going to be set by the circuit, so your purchase should be dictated by circuit design more than what opamp it comes with. As they say in realestate, you can fix your house, but you can't fix your location.
Then—after you have the amp in hand—you can start opamp rolling to see which sounds the best in your rig. This assumes that the opamps are socketed, of course. Or you can design it from the ground up for a particular opamp, but ... well, I think that's overboard, personally. I hope that was useful to someone... though it's probably riddled with half-truths and outright wrongnesses.
