I can weigh in on this now as a fan of a team that has done both now.
I think the regular season is the better accomplishment objectively. Winning the tournament is a more exciting thing though.
If the question is which would you rather experience? It would be the tournament champion. I mean, who wouldn't rather be 2nd or 3rd seed and win the tournament, vs being first seed and getting bounced?
However, if you come to me before the season and say, "You can choose, your team is either going to be the regular season champion or the tournament champion..." then I'm choosing regular season champion. That means you've had a hell of a year, won way more games than you've lost, and are almost certainly going into the tournament as a 1 or 2 seed. That's guaranteeing an entire season of great results to enjoy.
Now, if I'm a Duke fan that is pretty much expecting that the regular season is always going to be pretty good to great, then I might pick tournament champion, because you're not that worried about the regular season anyway.
But look at UNC...if they win the tournament this year they will love it...but would they trade that for all the painful losses this year?