Honestly, the HD 6XX sounds more linear, as in the bass, mids, and treble are all more in balance with each other than the PC37X, and the HD 6XX sounds more refined. The PC37X sounds more fun, with a tasteful boost in the bass and treble. I have an HD 650 (and you can see a link to my PC37X review in Massdrop’s product description), and I use them with a GSX 1000 external DSP/DAC/Amp, and both of them have good imaging and great benefits from the binaural surround processing (some headphones are not detailed enough, and smear the surround processing so much that they ruin the effect).
For gaming, I use the PC37X, its Optimized for that, and you don’t have extra wires and magnets to worry about with a mod mic. I think the Sennheiser microphone sounds better than the ModMic too, plus it’s easy to mute it when you feel a sneeze coming on or you want a bite to eat. My buddy uses an HD 600 with a modmic, and it clearly works for him. For music performance, with a strong amplifier, the HD 650 I have (which is like your HD 6XX) is obviously more linear, refined, and scales up more and more the better the music files/software/DAC/amp/etc you feed it with, but the PC37X is subjectively still able to offer up a ton of fun and objectively have performance that isn’t fatiguing.
The Schiit Magni is one amp option that has enough power to support the full frequency response of both headphones, and the Modi is a well-regarded DAC as well. I don’t know what you’re using for an ADC to plug in the microphones from either setup.
In summary, the PC37X costs less and can be more convenient, but the HD 6XX has an increased performance commensurate with the increased price. I think both are good choices.