I've tried to explain the power side of things in this post for the hd650 if that's what you're looking for:
http://www.head-fi.org/t/197776/sennheiser-hd650-impressions-thread/27645#post_11806577
keep in mind that all musics aren't recorded at 0db (full scale signal), also if you reduce the gain in an EQ, or use replay gain, the amp will need the to have the ability to compensate for that extra gain and still sound as loud.
aside from that, an amp with lower distortions will have lower distortions. one with better crosstalk or slew rate or whatever variable you will see measured, should manifest those differences. so at least measurable differences will exist. how that will be meaningful is for you to decide. one with crappy fidelity will be crap whatever the design if you look for fidelity, but it may feel superior to a transparent amp if the defects sound nice to you and you're really after something pleasing instead of hifi.
most pretend that they care about fidelity and then just buy whatever feels nice to them instead of what measures well. you even have such concepts as people calling transparent amps "boring". and some will simply pick whatever they like and call it fidelity often going as far as rejecting measurements or electrical laws when they don't align with their personal opinion of superiority.
so IMO, even before looking into electrical requirements and signal fidelity, one should make his own mind about what he's really looking for. and then avoid lying to himself(and others) about it. ultimately what matters is for you to enjoy your music, so there is almost no bad choices here, only bad excuses.
at a pure electrical level, a 300ohm headphone is in fact pretty easy to drive. you may have to put more thought into picking the an amp when using IEMs or planars, or speaker amps. that may feel hard to believe when you read the hd650's topic, but I really believe this to be true based on my understanding of electricity.
when comes the time to make a choice, of course other parameters will come into play. most will be pepsi vs coca cola kind of choices. where you're not even sure if it really matters or why you have a preference. starting with tube vs solid state. then within each group there actually are several sub groups of components and designs, each with some legitimate pros and cons. as we can't really expect to know everything about amp designs(I certainly don't), we usually end up picking something for the wrong reason. because we got convinced by someone as confident as he was ignorant. or because it looked good, or because a choice needed to be made anyway. the truth being that you can nowadays make very clean sounding devices using almost any technology so it doesn't matter all that much if you can find some measurements to make sure the amp you're looking at is more than a brand name and some eye candy.
tubes will run hot so depending where you live, I'd consider that an important variable^_^. also if you're looking for a colored amp(one that has it's own signature and distortions), tubes still seem like the logical choice.
price is also a BS factor in the sense that some components are expensive because they're more expensive to manufacture, not because they make a better sound. with the same idea, some amp manufacturers are in fact 1 or 2 guys in a room making amps from scratch. so obviously they have costs that are very high compared to some big brand making XXXX amps a day in a factory wherever it's cheaper.
so all in all, past electrical specs and personal taste(if you're lucky enough to be able to try the amps), everything else could just be random and it wouldn't be harder to pick.
else you can simply ask questions by mail to the manufacturer. within the audio industry, I'd go as far as saying that amp manufacturers feel the most honest/less dishonest of all(IMO!!!!!). if it's a giant brand like sony, don't waste your time the guy answering won't have a clue, but otherwise my personal experiences have almost always been informative, even when they deliberately avoid answering some questions, I find that it's also a meaningful intel.
I'm sure you would prefer that I told you "buy XXXXX, it's the best!" but I don't have any such opinion on the matter. so good luck ^_^.