As a Texas native and longtime former Austinite, I would not recommend UT if cultural/city experiences are important. Austin is a giant suburb. It’s more strip malls and highways than anything else. Big music fests, and that’s about it. The heat is miserable, there are no seasons, just summer and not summer. Nola’s heat is far more tolerable, and when it is intolerable, school is not in session.
Austin doesn’t have a southwest vibe to me at all. Southwest, to me, denotes connections to native/local cultural heritage and roots. Austin doesn’t really have much local/native culture that wasn’t really developed in the last 20 years. (In fact, that’s an issue bemoaned by many Austinites. What made Austin so Austin doesn’t really exist anymore, and what makes Austin what it is now didn’t arrive until the early 2000s.)
Interestingly, UT has a much more diverse student population than Tulane, while Austin is not nearly as diverse as NOLA and is heavily, heavily segregated.