Army Spec Ops preform more highly covert operations like doing deep reconnaissance , special direct action missions( like hunting down terrorist cells),guerilla warfare and counter terrorism operations.
All Army Spec ops are Airborne qualified as well.
Airborne Divisions are conventional. They are airborne qualified meaning they are qualified to be inserted into a combat scenario vis static line jump. These divisions are pretty much the same as regular army infantry divisions, except they are airborne qualified. They do the same things as regular infantry, which is to wage conventional war in a larger force. pretty much the troops you see on CNN patrolling the street are regular army or marine infantry units, so thats pretty much like an airborne division, except airborne divisions are airborne qualified, and all airborne divisions are rapidly deployable.
So in simple,
army spec ops- conduct smaller special covert ops, and are used a lot as an unconventional force.
army airborne divisions - conduct more large scale , conventional warfare operations much similar to regular army infantry divisions .
