What's the difference in closed, semi-open and open headphones? Which is best for Fractal?
Biggest difference is going to be in noise isolation. Open headphones have holes in the ear cups so it lets in outside noises (like your guitar pick on the strings or noisy kids in the house) and also lets some of the sound from the headphones out into the room (so you won't want to use these while recording with a sensitive mic). Closed back headphones will tend to seal off the sound much more, letting in much less noise from the outside world, which can be helpful if you are in a noisy environment or if you want to do things like pitch shift your guitar.
The other big difference is that some closed headphones can be more fatiguing as they can do things like create some pressure on your ears (if they have a really tight seal with good bass) or have reflections bouncing back into your ears. A good open back headphone can sound more natural, more like a set of speakers in the room. But this totally depends on the headphone and your preferences.
If you use headphones quite a bit, I think it's nice to have both options. I used open backs for a few years quite happily but right now I'm in a bigger wooden room that has more reflections so I have to turn up the headphones louder to drown out the sound. That's one reason I've been trying some more closed back headphones as it blocks out more outside sound and I can play at lower volumes.
On a budget (under $100), there's two really popular headphones. For closed back, the Sennheiser HD 280 Pro is very popular, it's been a studio standard for many years for tracking. For open back, the Philips SPH 9600 has been a really well regarded budget open back. It'd start there...you can order both and either keep both or return what you don't like.
For a little more money, a lot of people like the Beyer headphones (DT 770, 880, and 990). These are really good German made headphones for the price, although they all have a treble boost. If you like the sound, they are hard to beat, and they have closed, open, and semi open models. I like the 990 most of all three but again for guitar it's on the bright side. If you plan to use just with the Fractal the 250 ohm should be fine, but the 80 ohm is also a good pick as it's a little easier to drive and can be used with more equipment. I would avoid the 600 ohm as they need a lot of power to drive, and avoid the 32 and 16 ohm versions as those are more consumer headphones that are reported to have less clarity.
And the headphones that most people really love under $500 are the Sennheser HD6__ series. The HD 600, 650, and 660 are all pretty well loved as very flat open back headphones (with some highs and lows rolled off a bit). The HD6xx from Drop.com is basically a rebranded HD 650 at $220 which is an excellent value, and why you'll see them EVERYWHERE in audiophile groups. Those have been my main headphones the last couple years.
Again, I've been trying a few different closed back headphones and I'm VERY impressed with the Shure SRH 840. It was creaky out of the box (I had to lube all the contact points), and it's a bit heavy and uncomfortable so won't be ideal for hours of extended use. But the sound seems pretty dang flat to my ears with a little more bass and treble that I don't get on the Sennheiser. I tried a few different ear pads on them (you can get replacements for $15-25), and I have a set of perforated leather pads with thicker memory foam compared to the stock pads. This has REALLY helped with the comfort as it relieves a lot of pressure on my head and keeps my ears from touching the headphones.
I'd be curious to try the higher end models (like the 940 and above), but at twice the price and with leaner bass I feel like it would be a side step. Other higher end headphones I'd like to try are the Beyer DT1770 and DT1990 which are the higher end closed and open back heapdhones that are supposed to tame the high end spike a little more.
As a follow-up to my headphone trial, I did return the Beyer DT 770 as it's too bright, and I ordered another set of lightly used Audio Technica ATH-M50X to test those. Most likely I'll probably use those for listening to music, so long as they aren't overly scooped. I kind of like the KRK 8400 but they just have too much of a bass boost and mid scoop compared to my other headphones, so those will likely get returned next week.