Wait, you have a set of headphones for your different moods?
Yes, but not all of them. Not listed: various in-ear and headband phones.
Can't you just have one pair of headphones?
Yes/No. I could buy one really nice set of headphones but I don't feel like investing $500+ into headphones that once warranty gives out/shit happens are done and you're screwed, or get lost or stolen. There's also the point of diminishing returns. The difference between $500 and $3000 headphones is much less than the difference between $30 and $200 headphones, although I'm sure some audiophiles would fight me for saying so. Different headphones are like different cars. Sometimes I want one that is powerful, at other times I require sensitivity and nimbleness, sometimes I want one that is able to get down and dirty. I don't really wish to compare headphones to people, but the characteristics are so completely different that you can't just tell people/headphones to act the same and get the same results. Which leads into your next question...
Do these really differ so much in sound? Is it even possible to have such difference in headphone output when they are all good quality?
Yes. For starters, there's bass/mids/high - every headphone has a certain balance between the sound output. Some excel in bass, which makes them good for electronic dance music, hiphop, R&B. Others have tight, strong highs, which is fantastic for vocals and rock music with guitars. There are closed-back and open-back headphones. Closed headphones are typically heavily padded and have thick shells to give good sound isolation, meaning that the sound is kept in and sounds outside are kept out. This is good for producing music, mixing, and DJing. However, this makes the sound feel "closed" and somewhat unnatural, so on the opposite end, open-backed headphones have a tiny or nonexistent light foam padding and you can generally look inside and see the headphone's guts. This simulates how you naturally hear noise from more directions, but unfortunately bleeds your music out and allows sound in. There's Circumaural headphones of which the pads cover the entire ear (tend to be closed back), and Supra-aural which sits on top of the ear. There's comfort and design issues. Everyone has a different taste in sound and music and therefore people will want different headphones.
Going back to the comparison - Say we take 100 different people who are fantastic singers, and tell them to sing the same song, you'll get 100 different variations.
So can't you fix that with just EQ balancing? Somewhat.
EQ balancing is done by warping the strength of the bass/mid/high signals. To use more metaphors, this is done as through "software", but the overall output strength is still very much under the jurisdiction of the "hardware". No amount of software programming is ever going to be able to allow hardware to do something that is not meant or powerful enough to do, although you can make an effort. You can't force a classically trained opera singer to sing metal "well", as it is also difficult to get a rapper to sing cutesy pop music. Grados headphones are renown for rock music and an unique, characteristic high-pitch treble strength (disliked/causes pain for some, others swear by it) that can bring out nuances in voices that you've never heard prior in your music, but I wouldn't want it to blast dance music at me.
Some headphones are good, some bad, some excel in certain areas and genres, while others are balanced (used for composition/monitoring) but unexciting to listen to music on when you really want to get into the music. And some are just plain shit and/or overpriced. *cough* skullcandy/ipod standard earbuds/bose *cough*
I know you audiophiles are going to hate me for not knowing that
I'm not really an audiophile as these are all simplistic explanations meant to educate- and it takes a certain level of pretentiousness to call oneself an audiophile, but I do absolutely love music. I'll also be honest here and point out -
good music saved my life.
EDIT: And once you find that first magical set of great music equipment that inspires love, it's impossible to go back to the tripe.