Let’s start with what’s not different: For all intents and purposes, build quality is identical. They take the same cables. Both do a good job with every style of music. Whichever you pick, you’ll be happy.
Don’t stress about making a wrong choice, because there’s not one. If a wine analogy helps, this is a bit like asking whether a Pinot is better than a Cabernet—the right answer depends first and foremost on your preferences, and your answer may change with the meal you’re pairing. Nevertheless, a good wine is a good wine, and both of these headphones are good in that sort of way.
A few months ago I decided to buy either a 58x or 6xx for a longtime friend as a gift, and I sat down to do some A-B listening between the two. Without getting into esoteric audiophile terminology, the overall presentation of the 58x feels more like swing music—vibrant, lively, in-your-face—whereas the 6xx feel more like lounge—chill, sultry, refined.
My friend is a metal-head and guitar player, and listening to some of his favorites, guitar riffs were a bit more forward and detailed on the 58x. I got him those, because I figured they’re probably a better overall match for his favorite listening: diving into a wall of sound and rocking out.
Meanwhile, I like all types of music, and I enjoy taking in the overall “big picture” of tracks—analyzing layers of instruments and taking in how all the various parts interplay with one another. The 6xx sound is a bit more open; I prefer them over the 58x for the ease with which I can indulge my own wanderings through chill tracks. But I prefer the 58x for things like intimate vocals, heavy rock, maybe EDM.
Back to the wine analogy, it takes a certain level of snobbery before a particular wine varietal makes any difference whatsoever to your enjoyment of a meal, and that only develops after plenty of experience, gobs of over-analysis, and a healthy dose of lost perspective. Since you’re coming from a place of inexperience, just pick a flavor—jam or chill—and start enjoying. Get familiar, then come back later and sample something else if you like.