Short answer: One is still used as a tactically viable way of getting troops into the fray and the other is more ceremonial.
Benjamin Franklin once said "Where is the prince who can afford to cover his country with troops for its defense, so that ten thousand men descending from the clouds might not, in many places, do an infinite deal of mischief before a force could be brought together to repel them?"
Both of these troops fit that bill over two hundred years later.
Out of all of the current military rivalries, this one still ranks pretty high on the list. As someone who's Air Assault and let his personal rivalry simmer a bit, there's no reason to keep it up. The differences between the two just keeps growing with each conflict.
(U.S. Army photo by Master Sgt. Sean McCollum, 29th Infantry Division Public Affairs)
By World War II, many forces developed their own form of Airborne infantry that soared into combat. Allied forces captivated folks back home with the tales of jumping into the European theater. Over the years, airborne operations can be performed in essentially two ways: static jumps (think of the age-old cadence "Stand up, Hook up, Shuffle to the door! Jump right out on the count of Four!") and HALO/HAHO, or High Altitude, Low Opening and High Opening (free-falling).
Air Assault rose in the Cold War and became more prominent in the Vietnam War. There are usually two means for getting troops into combat, FRIES, or Fast Rope Insertion/Extraction, where you grab a piece of rope and slide out of a hovering helicopter and just Air Insertion, where the helicopter lands on the ground and troops hop out. Technically, there's also Sling Load operations, where you attach things underneath a helicopter, but that's more of a special task that's assigned to Air Assault qualified troops.
There's several more ways of leaving a helicopter. Like SPIES and Helocasting, as seen above (U.S. Army Photo by Sgt. Codie Mendenhall)
But in the wars since 9/11, you can count on one hand the number of combat jumps performed by US troops. They were done twice in Iraq and once in Afghanistan — and all three to command and control airfields.
Making a combat jump authorizes you to wear a Combat Jump Device. It's a gold star that adorns the Parachutist Badge and is often referred to as a "mustard stain." Finding one of these bad asses outside of Jump School is like finding a CW5 — you know they have to exist somewhere because you've seen the badges at the PX, but it still sounds as plausible as any other barracks rumor.
There isn't as comprehensive list on total Air Assault missions because it's far more common. It's just another way to get around.
Many combat arms guys can tell you that they never went to Air Assault school, but still do Air Assault operations in country. The only Air Assault task restricted to someone who actually went to the school is the previously mentioned sling load operations. Even that has its "volun-told" feel to it. Sling loading has a risk to it that could be deadly if not done properly. Only Airborne school qualified personnel are allowed to complete airborne jumps (because of the weeks they spend just learning how to fall properly).
(U.S. Army photo by Visual Information Specialist Jason Johnston)
Sure. We have our disagreements and will probably flame each other in the comment section. They're both ways to get men out of a perfectly good aircraft.
We both deal with a heavy amount of prop / rotor wash that training can never prepare you for. And both of our badges are still highly sought after by badge-hunters — usually a staff lieutenant or junior NCO. And they both will probably correct you by saying "well actually, according to Army regulation..."
Wear your blood wings proud, my brothers and sisters.