The major difference between men's gymnastics and women's gymnastics is in the events, or gymnastics apparatus, on which the gymnasts compete. They only share two events in common: vault and floor.
Female gymnasts compete on four events total: vault, uneven bars, balance beam, and floor exercise.
Males compete on six events, and compete the events in a different order: floor, pommel horse, rings, vault, parallel bars, and high bar.
The Differences on Floor Exercise
Both male and female gymnasts compete on the same floor exercise mat, but the women compete to music, while the men do not. There are other rule variations as well. In general, dance moves such as leaps and jumps are part of the requirements and scoring on women's floor but not on men's, and men are required to do more tumbling skills overall. Men typically perform harder tumbling passes. Women used to be able to perform a lunge at the end of a tumbling pass, but as of the 2012 Code of Points, women are now required to stick tumbling passes.
The Differences on Vault
Women and men both perform on the same vaulting table, though the men usually have the table at a higher height than the women. The vaults performed are similar as well -- in general, men perform more difficult vaults than women. The top male vaulters often perform double-flipping vaults, such as the handspring double front and Tsukahara double-back. Very few women have attempted these, even in training.
The men and women used to compete on a vaulting horse -- and men vaulted over it lengthwise while women vaulted widthwise -- but the horse was replaced by the table in 2001, mostly for safety reasons. The table is considered a safer alternative to the horse, with less chance that the gymnast will miss the table (especially during Yurchenko vaults) and suffer a severe injury.
Uneven Bars vs. Parallel Bars vs. High Bar
The uneven bars (a women's event) and parallel bars and high bar (men's events) are all different from each other as well. The uneven bars and parallel bars are usually made out of fiberglass and are bigger in diameter, while the high bar is made out of metal and is smaller in diameter. (Therefore, gymnasts' hand grips are different for the different type of bars, and it's dangerous to use the wrong type of grip.)
The bars are also set up differently. The high bar is a single bar about 9 ft. from the floor. The uneven bars are two sets of bars, that run about 6 ft. apart from each other and stand at about 5 and a 1/2 ft. and 8 ft. high. Finally, the parallel bars are two bars that are only about a foot and a half apart, and about 6 and a 1/2 ft. off the floor. (All heights are adjustable, though some are standardized in Olympic competition.)
The Competition Format of Men's and Women's Gymnastics
Both men's gymnastics and women's gymnastics (technically called men's artistic gymnastics and women's artistic gymnastics) have the same basic competition formats in the Olympics. In 2012, five gymnasts were on a team, with four gymnasts competing on each event in preliminaries and three gymnasts competing on each event in finals.
Gymnasts qualify into the individual all-around and event finals based on their qualifying scores, and 24 gymnasts make the all-around, eight into each individual event. Only two per country can qualify into each specific final however. All of these rules are standard across men's and women's competition.