The D1 appears to have more power, and optical inputs are a nice thing to have for connecting to consoles, or optionally, using with your desktop that has an optical out. It might work. Again, I'd do some research because I don't own a D1 or really any audio engine product, but from what I've read any DAC/AMPs that solely rely on USB power generally can't drive 300 Ohm headphones very well.. Currently, I have an FX Audio DAC X6 that should power the 6XXs, though, admittedly, the dac, and the amp isn't amazing, it's pretty affordable.
It really depends what you'd like to do. The hobby is such that you can go as crazy as you want and/or as your budget allows. Sound is very subjective after all. I don't want to tell you absolutely go for a product I've never touched to power another one I don't have yet lol. But I can give you options.
The HD600 Series seems to like balanced as well. Which can be done via 4-pin XLR like on the Schiit Jotunheim, or 2.5mm TRRS like on the Fiio Q1 Mark II.
Again, I don't have personal experience, but I do recommend watching some videos, etc. yourself to make your own decisions. I've heard good things about the Sanskrit 6th Gen DAC from SMSL, and the Modi 2 Uber from Schiit, as well as the Aune X1s, which is the reason I joined that drop. I'm hoping the Anniversary Edition of the X1s has enough extra power for the 6XX.
If you have a local store that would let you try some 650's, especially on some different setups, or a friend with those things, that would be the ideal way to learn what sounds best to you. I would generally say go about cheaper than the D1's retail price for a first time Amp/DAC just because you don't know if it will be powerful enough for the 6XX's, though it looks like it'll be enough for most other headphones.
Sorry man, it's late and I'm sorta rambling. I'm sure someone else will chime in and let us both know whats up. Good luck