If you are choosing between a business or first MileSAAver or an economy AAnytime, the only reason to choose AAnytime is to maximize your flexibility in changing flights. An AAnytime award can be changed essentially any time until the day of departure with no penalty, provided there is a seat available, whereas to change a MileSAAver flight, there would need to be MileSAAver availability on the new flight and you would be assessed a $150 change fee.
The incidental costs and benefits at the airport would be roughly the same. But flying in the premium cabin means a better in-flight experience, and on intercontinental routes you would also be comped access to the Admirals Club.
AAnytime awards are generally treated as full-fare (Y) tickets. An AAnytime economy ticket, for example, will include two free checked bags and a Priority AAccess boarding pass. You can also request seats in the Main Cabin Extra seats or Preferred Seats for no charge. There are no fees for any changes to your flight, although there is a fee for redeposit (unless they are from an account with AAdvantage Executive Platinum status).
They differ in that award tickets of any kind cannot be endorsed, do not earn miles or Q-miles, and cannot be upgraded.
MileSAAver awards have all the above restrictions and lack all the benefits, and are subject to tight capacity controls to boot. They do incur a change fee if you alter your routing (again, unless the miles come from an Executive Platinum account), though not if you keep the same routing and merely change your travel dates. A MileSAAver economy ticket is like a discount economy ticket; it does not get the baggage and priority boarding benefits.
But as you have found, a MileSAAver business or domestic first ticket costs the same number of AAdvantage miles as an AAnytime economy ticket. Flying in a premium cabin also gets you a baggage allowance and Priority AAccess, a priority checkin line at the airport, plus an invitation to the airport club if you are on an intercontinental flight, not to mention a seat that is going to be at least as comfortable as any Preferred Seat in economy. Given my travel patterns, I will always opt for a restricted premium cabin ticket over an unrestricted main cabin ticket when they cost the same.