Which is the best, Air Assault or Airborne? Which one is the best for a career?

Air Assault is just a quick class, but Airborne is a way of life. Which means that one is a badge and the other is a unit that you serve at for at least three years. The distinction is important because it means serving in an Airborne unit for several years. As you may already know, there are reenlistment bonuses for Airborne positions, extra points for serving in an Airborne position for promotion to E5 and E6, and on the senior enlisted boards in almost every MOS, Soldiers in Airborne units are usually selected for promotion above their non-Airborne peers.

Airborne is an SQI, like Drill Sergeant or Recruiter. It's a qualification needed to serve in a specific role. Air Assault is an ASI, it's an additional skill that enhances your current position, like Battle Staff, Master Fitness Trainer, JTAC, or Pathfinder.

Gaining a skill is always looked at favorably by the promotion board, but serving in a more demanding position (like an airborne unit) is always looked at even more favorably

What? You trying to start a riot???
Of course AIRBORNE!!

It all depends on what YOU want to do in your career. In the end they are both qualification badges. You go airborne you can be assigned to airborne units like 82nd, SF, and other airborne brigades. Air assault you get if you get assigned to 101st. Me- I would chose airborne every day of the week and twice on Sunday.

There it's no true "best" between the two. They have specific purposes they are used for and they fill that function well. Career-wise, Airborne will allow you to to be assigned to many other units in the Army, whose missions could interest you more. For instance, an Airborne unit can and does conduct Air Assault operations. However, an Air Assault unit cannot conduct Airborne operations, unless specific portions of the unit have been officially designated as Airborne.

Do whichever floats your boat though. The rivalry between Airborne and Air Assault is purely endearing. At the end of the day, people with special/additional skills help the Army succeed in its mission, and that's what really matters.