Talk about a a can 'o worms. VERY broad subject.
Bottom line - YES: most voices can sound better when recorded well...from a little better, to a
LOOOT.
I'd argue that MOST singers sound better recorded than 'natural'.
There's a few people who sing so beautifully that, heard 'live'/un-mic'd, is still beautiful.
But even those...and especially the ones who don't sound so great (yeah, I'm in that bunch) can be made to sound a lot better, when properly recorded.
A decent mic; a good room; proper use of EQ and/or comps'n; judicious use of FX; and even a light touch of auto-tune/pitch correction can all make a big difference.
"What's the difference"? Lotsa things (see above)...I guess I'd venture to say that when listening to someone 'natural' (no mic, recording, processing, etc) is going to give you a better sense of what they
actually sound like - the tone, inflection, breathing, pitch control, dynamics. But recording gives you a lot of control over a lot of things.
And I don't haveta tell you that MOST vocal recordings - pretty much everything in the last couple decades - bear practically no resemblance to the singers' natural voice. I was watching a vid someone here posted about doing vocal production in the studio, and it featured a guy who'd engineered a big country-music (I use the term loosely) hit for some woman singer. It was re-DONK-yoo-luss...he took the vocal she sang, copied it about 5 times, treated each one with a dizzying array of eq'g, various FX, etc etc, and mixed them all together...it sounded great when done, but sounded NOT ONE BIT like the woman's voice as it was recorded.
There's my $0.02...