as 03USMC said, both divisions are light infantry. The 82nd has around 14,000 troops while the 101st has over 20,000 soldiers. I know the 101st is in the process of making a new brigade, and I am sure the 82nd will follow suit.
The main difference between the divisions is Air Assault vs Airborne. The 101st sole purpose is to do air assault operations while the 82nd does mainly airborne operations. I know the 82nd can air assault too, but only in a limited capacity. Being Air Assault is why the 101st is much bigger than the 82nd. Here at the 101st, we have choppers, choppers and more choppers. I believe we got over 300 of the birds.
THe 82nd and 101st share simular unit histories. The 82nd Division was first formed during WWI. The division was reformed in WWII and served in Italy and in Europe. The division was reformed in the 50's. the 82nd played a limited roll in Vietnam. Troops from the 82nd were sent to the Dominican Republic and to Vietnam as rapid deployment troops during the TET Offensive.
The 101st was formed during WWII. THe 101st served in Europe. The 187 INF REG was added to the 101st later. The Rakkasans were the first US paratroopers to jump against the Japanese. The division was reinstated in the 50's and served in Vietnam. The 101st did the main fighting in the A Shau Valley (Movie Hamburger Hill).
After Nam, most should know the histories of the 82nd and 101st.
edit: oops I forgot the 82nd did a jump in Korea. There was a general in Korea that saved the 82nd from being used in suicide missions. GEN Ridgway took over the ailing 8th Army and turned it into a bunch of fighting men. After the successful airborne operation, higherups wanted to use the 82nd for impossible missions. Since Ridgway commanded the 18th Airborne Corps in WWII, he knew better and protected the 82nd.
"The best form of taking care of troops is first-class training, for this saves unnecessary casualties." Erwin Rommel