You'll still get your luggage allowance, boarding priority, and seat selection (if saver seats are available at time of booking - if not, you'll be assigned one at boarding, possibly in Premium - extra legroom, free drinks, if not all assigned). AS lounge at LAX is pretty small compared to some of their others (much smaller compared to AA flagship), and is pretty limited in terms of complimentary snacks (soup, salad, fruit, cheese, pancakes, breads, snack mix, etc) and beverages available - a few wines, decent draft beers, mid-tier liquor drinks, barista bar. More substantial food, better wines, and premium liquors are available at a cost.

I've not visited the AA flagship lounge, but from browsing reviews it looks to be much nicer, and if you're connecting from an international flight, it has the benefit of shower suites. Only benefit for AS lounge would be it's close proximity to your departure gate, where AA will require a 15ish min walk through airside underground tunnels.