I've come a bit late to this, but it's an interesting area, and one that I've never really thought of. I think I have three types of solos that I play:
1. Full on, make it up as I go along,improv. I mainly do this when I play at home, originally with guitar, now with sax as well. Stick on a backing track (one of mine or from youtube) and just play for hours (well, it used to be for hours, now I've got to fight to find the time). Also fun when I'm out at bluesy-type jams, or we're freewheeling at band rehearsal, and with some cover songs...
2. In my cover band I might do either full improv, or partial improv around a theme, or note for note soloing, depending on the song. So, for example, at tomorrow's gig I shall be playing (on guitar or sax as appropriate) the little lead in American Idiot note for note ('cos it's part of the song's structure), as I will with No-one Knows, Dreaming of You, 7, and a few others. I'll riff around a theme on the Riverboat song, Rio, Nightboat to Cairo. I'll will make up as I go along on Rocks, Seven nation army, Are you gonna be my girl. So it depends on the song, and what (I think) audiences expect to hear.
3. In the Floyd tribute, I play the sax solos as close to note-for-note as I can, because they are pretty iconic. The band as a whole tend to play live versions of songs rather than replicating studio albums exactly, and I can see the value in that, but for me the sax has to be spot on, because that's what I've listened to over the years.
There's a comment above about cover bands (or tributes) playing things note for note correct, otherwise it might seem they can't play the song properly. I tend to agree on one level (which is why I always learn solos the way they were originally played, then either keep that or tweak it), but then again I've heard lots of bands live who don't stick to their studio versions, so there is room to change things round.
Anyway, that's my rambling way of saying, a bit of everything, context dependent