1.02 is simply harder to crash. The infamous "Black Hole Glitch" (involving Peach, Ice Climbers, and a food item) was patched, and the game handles extended play sessions on console better without memory leaks.
While the differences between 1.01 and 1.02 are subtle, they are critical at high level. Here are the most famous changes found exclusively in Melee 1.02: melee 1.02 iso
In versions 1.00 and 1.01, the Ice Climbers' Nana (the secondary Climber) behaved unpredictably. 1.02 introduced the consistent desync mechanics that allow players like Slug and Army to perform "Hand-Off" chaingrabs. Here are the most famous changes found exclusively
Release shortly after 1.00, this version fixed several game-breaking bugs but still contains many of the "advanced techniques" in a slightly different state than 1.02. It is uncommon but not as rare as 1.00. It is uncommon but not as rare as 1
A: No. There is a common myth about a "1.03" or "Rev B." These do not exist for NTSC. All 1.02 discs are functionally identical. Only the Japanese version has a "1.02" that actually differs slightly (character portraits), but competitive players do not use it.