Both are fine schools and will train you well. If you are in state, I would choose LSU unless you receive a sizable scholarship to Tulane or there is a substantial reason not to. A few notes from my time in New Orleans:
1. Tulane's class is largely out of state, just like their undergrad student body. This can be a plus or minus. LSU is almost entirely instate, but now accept around 10-20 out of state.
2. Tulane sends more grads to out of state residency programs based on their match lists I've seen. I think this is largely a function of the above. Out of state students probably want to do residency closer to home, and possibly set themselves up for practice outside of Louisiana. That said, many Tulane grads end up staying in state, but not to the extent LSU grads do.
3. Tulane is pass/fail for preclinical (to my knowledge), LSU does H/HP/P/F.
4. The plant and facilities at LSU are nicer from what I have seen, but Tulane is very much involved in the new hospital. This really shouldn't be a deciding factor.
5. From what I understand, LSU has a well regarded ENT department. I know first hand the department is very open to interested students.
6. Both schools are very involved in community outreach and service.
Hope that helps. I am an LSU student and have greatly enjoyed my time here. I feel that LSU is likely a slightly more "traditional" school, while Tulane is a little more unique among schools in the south. Good luck next year, both are great schools and you'll have a great time either way.