Elite status means more perks from your program. These perks vary by program and also be level of elite status within each program (typically 3 or 4 or more levels as well as the "no status" level).

Some examples of status perks (not necessarily relevant to AAdvantage or any specific program and status level).

Access to lounge.
Extra baggage allowance.
Special check-in queue.
Priority waitlist.
Priority for upgrades.
Vouchers for upgrades, awards, hotel stays, car rentals, etc.
Preferential seating.
etc, etc.