Only as important as the creator makes it. It shouldn't ever restrict song choices, but it's a fun thing to have for roleplaying/what-have-you. I'm kind of with Myst in that it's fun to have to imagine scenarios with or roleplay or write stories, but beyond that, I don't think it should be taken too seriously. My UTAU have personalities in my head that don't always line up with how other people depict them, and that doesn't really bother me?
Like, Halt gets interpreted a lot of different ways--as a dork who acts like she's smooth, as a huge nerd who watches bad anime, as a tough independent punk type, as a shy and nervous awkward turtle, as a total ladykiller or domme... but to me, she's kind of different. She's like a bit of myself squished together with... idk, something. To me, Halt is a level-headed big-sister type who cares about her brother and adoptive sister even if she sometimes throws her shoes at them when they won't behave. She's a huge nerd who can't cook so she eats a lot of Chinese takeout, goes to bars on the weekends and sometimes gets with someone but never really falls in love. She throws girly sleepovers and makes terrible jokes and wastes a lot of time on the internet. She works hard but takes life easy, because she's just very content and laid-back, y'know? I dunno. I guess the point of that rambling was that I see her as someone very different from how others do and that's fine with me?
If I'm going to even consider personality in an UTAU, I like to either see it be free for interpretation or moderately interesting. Simple, stereotypical things like "tsundere" or "shy and quiet" aren't interesting. I like to see at least a little bit of depth of character if they're going to have a personality, but then again maybe that's left over from my roleplaying days ahahaha...