American Airlines has 3 elite status levels, Gold, Platinum, Platinum Pro, and Executive Platinum.   Here is American Airlines description of status levels. 

The Professor gives his viewpoints of the advantages of having Platinum status:

Free seat selection in coach:   For The Professor, this is the primary advantage.  It is always preferable to be in the front of the plane.  Being in front guarantees that exiting the flight will be quick, which is especially important when needing to make a connection. 

Preferred Seating: American Airlines has seats available at no extra cost, called "Main Cabin Extra" to Platinum members that offer just a little more legroom than other seats on the plane.  And because of free seat selection, it is easy to switch seats online and possibly get an empty row or next to an empty adjacent seat. 

Priority Boarding:  Another very useful perk is getting on the plane before most other travelers.  There is always room in the overhead bins. 

Mileage Bonus:  Not a great perk, but more miles add up for more mileage award seats. 

Upgrades:  In general, the higher the elite status, the more eligible a flyer is for an upgrade to domestic first-class seats.   On average, The Professor is upgraded 40% of the time on domestic flights over the past 2 years.  Upgrades on international flights (other than Canada) require either award mile redemptions or cash payments and Platinum status is not an advantage in getting international upgrades. Free upgrades are very rarely granted when flying on an award flight to any destination.

Reduced or Eliminated Baggage Fees:  With his elite status, The Professor has never paid for baggage on American Airlines.  However, The Professor rarely checks bags.

Platinum Telephone Number:  Also, not a big perk since waiting times on this "special" phone number can be long too.

Since The Professor finds good scheduling (home airports are Dallas DFW, Charlotte CLT, and New York JFK) on American, it is worthwhile to maintain elite status on American Airlines.