New Orleans has a MUCH larger bar/club/entertainment/restaurant scene than Baton Rouge. However, if your BF wants to attend LSU then living in New Orleans is not really practical.
In Baton Rouge, most of the crime is in the northern part of the city ("North Baton Rouge"). Once you are south of I-10/I-12 there is little crime. There are dozens (maybe scores) of apartment complexes which are down river from the LSU campus that cater to faculty, staff, and students at LSU. You may or may not qualify for on campus housing at LSU but you should check on that.
New Orleans is more complicated due to the way the city developed and with a few exceptions good & bad neighborhoods alternate. Jefferson Parish, which is just up river from New Orleans is generally safer but is stereotypical suburban America.
People in both cities are friendly, and with regard to rents you get what you pay for.
If it were me and I was not planning to attend LSU then I would live in New Orleans because it is home. I would try to live in Lakeview, the Bayou Saint John/City Park neighborhood, or the Warehouse District (which the City calls the Arts & Museum District) . If I could afford a place inside the French Quarter with reserved off-street parking then I would probably choose there.
https://www.neworleans.com/plan/neighborhoods/
https://www.neworleans.com/plan/neighborhoods/lakeview/
https://www.neworleans.com/blog/post/guide-to-bayou-saint-john/
https://liveatesplanade.com/ (this is an apartment building on the Bayou)
https://www.neworleans.com/plan/neighborhoods/arts-warehouse-district/
https://www.neworleans.com/plan/neighborhoods/french-quarter/
Pensacola is about four hours drive from New Orleans five from Baton Rouge and I recommend you visit before deciding.
You are welcome to write to me via Yahoo Answers if you want more info. I don't work in the real estate industry so won't try to sell anything to you. In the Travel category I am the Top Answerer in the New Orleans section, third in the US section, and fifth in the world, so I am not a creep.
Best wishes!