I have tried this both ways that you suggest.
Here's what I found:
With a band, all playing together means the 'guts' of the song usually comes across better. We play rock and groovy pop + indie, so a performance with some mojo (intangible and yet so important) is really what we're after. This works best as a band.
Recording things individually can work but is more time-consuming: If you have 1 x track for each instrument (drums = 1 instrument although it might be over several tracks), then the time taken to record, say, a 4 minoute song is 4 x 5 instruments (in your case) + any time for vocals. This becomes significant if you're recording a whole album.
If recording vocals live in the studio while playing, it's quite difficult, even impossible to get good separation from the instruments, especially for the drummer. you end up with bass and drum bleeding all over the vocal mics.
While recording my own stuff, so it's just me as a musician, obviously I have to play everything individually. It's GREAT for experimenting because you can just wipe a track and try again, without anyone else having to wait about or play the sng with you again. However I've found it's a bit harder to get the 'glue' that makes a great sounding rock band.
If recording everything individually, remember that someone has to go "first".. drums, I guess? Then the bass, but they have to imagine the rest of the band with them because it hasn't happened yet. For both of these instruments, it's a tough call to get an emotive performance from such an empty space. (Edit: To get around this, you could record a 'pilot track' just for a guide, which doesn't make it into the final mix- although I think that might be confusing and is yet more recording. It'd be better than nothing though. Thanks for the comment, Richard Mtzler)
With my band, we've found the best combination is to record the 'backline' (drums bass guitars) all in one go (but of course on separate tracks), then add the vocals later so that you get good separation and of course you get several chances at a good take. I'd suggest this as a way forward for a band scenario.
I've referred to that 'intangible Mojo' and 'guts' when playing (effectively) live: It's hard to define this as it may be change in different scenarios, but there's an element of live interaction that happens when musicicans play together and can communicate visually, or react quickly to hearing whatever else is going on.
It's related, I think, to the timing of the drums and bass (a human charactarestic which might not go out of time with a metronome, but might subconciously accentuate parts of the music)- or if you're not using a metronome, the tempo may change which adds to the tension of a tune. If this sounds odd, have a listen to Brown Sugar by the Stones and clock the tempo change after the intro- also Honky Tonk Women, and Common People by Pulp. Doens't harm the song, in fact it adds to it - that's definitely some humans in a room getting excitable about what they're playing :-)
There's also the sneaky licks and tweaks that might be added at whim while playing together, but when recording just that single instrument, perhaps the inspiraiton to do this isn't there, or is different.