Headphones and speakers present sound in very different ways: speakers "play" the room, headphones "play" your ears. To clarify, when you're listening to speakers, more sound is reflected off the floor, walls, and ceiling than there is direct sound coming from the speakers. Which means your room's acoustics have a major effect on the sound, and no matter how good your speakers are your room alters their sound. Rooms with really good acoustics in homes are extremely rare.
Headphones directly inject sound into your ears, so the room's influence is completely taken out of the equation. Ah, but the shape and contours of your outer ear, and your ear canals affect the headphone's sound, but to a lesser extent than your room's acoustics change your speakers' sound.
That's why you can hear even the most subtle details of a performance like the singer's breaths, fingers sliding over strings, or the recording venue's acoustics over a decent set of headphones; details like those get lost over speakers. That's what I mean by "accuracy"; those details are present in the recording, but lost to some degree in your room's reflections when played over speakers. As for enjoying the sound of speakers or headphones, that's a subjective call.
Another major advantage for headphones is most full-size models use a single "full-range" driver in each earcup. Speakers divide the frequencies into -- bass, midrange and treble, or bass and treble -- and play each frequency band over separate drivers, and losses inevitably occur. Full-size headphones suffer no such losses.
When it comes to bass, speakers outperform headphones. While headphones can reproduce deep bass accurately, headphones never let you feel the bass the way you can with speakers. Big speakers can energize an entire room with bass, you feel in your body and through your feet. Headphone bass is all in your head.
Speakers stereo imaging is also more realistic, more like what you hear in real life than even the best headphones. That said, headphones, especially open-back, over-the-ear headphones can produce a stereo image that can sound like it's coming from outside your head.
When I posed the speaker vs. accuracy question to my Facebook friends one guy hit the nail on the head and said. "Oh, this is easy... headphones are more accurate, speakers are more realistic." Then a number of engineers chimed-in, ones that used headphones when recording, editing, and mixing at least some of the time for the accuracy, but many other engineers relied exclusively on speakers. A few mastering engineers used headphones some of the time.
The speakers vs. headphones debate will continue to rage, but one thing is certain, a truly great set of audiophile headphones are a lot more affordable than a great pair of audiophile speakers. You'd have to spend many times more on speakers and amplifiers than headphones to get even close to the same resolution of detail.
In the end it's not an either/or question, ideally audiophiles or anyone who really loves music should have quality speakers and headphones.