Which of these would have better sound quality: Shure SE215 or ATH M40x?

Since you placed so much importance in audio quality that will be my first response, but not precisely on quality because that's subjective but of audio signature. The SE215 is BASS-BOOSTED for people who enjoy bass vs the ATH, according to one reviewer, tuned TUNED FLAT. You may not have an idea what these means if you have little experience with headphones. A short explanation if you find yourself reaching for the bass knob all the time you probably like the SE215 better, but if you enjoy quiet, classical, look for a particular instrument in a seas of instruments then ATH would probably be better.

Now my God one is an In-Ear and the other is Over-the-Ears. That's a huge thing. One man's opinion, I stopped using over the ears once I wore my first pairs of In-Ears. I find over-the-ears not for long wearing, they make my earlobes hot, then any movement, and the sound may change and the noise the cable, the band, the cups make annoy the heck out of me. I love the In-Ear, oftenly called IEM the best thing since slice bread. I love the compactness, go-anywhere, this may not matter to you, but any movement on my part, no problem, and I love the noise blocking. I am in my world and literally tune the world out, and without turning it up at all, when IEM are inserted properly you get insane bass, because the sound has no where to go but directly into your ears. Warning though because IEM are earphone+earPLUGS in one, some people don't like the feeling of wearing earplugs, and for this there is no way to describe it in words until your first pair of IEM.

And that's all I have to say.