Super Mario 64 is most notable for essentially laying the groundwork of modern 3D video games. The team behind the title started development by focusing almost entirely on how Mario moves. In doing so, they created one of his most flexible movesets.
Of course, there’s the usual running and jumping of any platformer. You can also punch, jump kick, long jump, back flip, ground pound, wall jump and dive forward. Or you can get into some of the lesser used moves, like a sweeping drop kick or sliding kick.
I’d even argue that tip-toeing is a unique element to this entry. You can do it in other Mario games, but rarely is it a gameplay element. Admittedly, aside from sneaking past sleeping foes, it’s hardly used in Super Mario 64. It was likely included to show off the new analog stick on the Nintendo 64 controller.
With each new 3D entry, Mario loses more and more of his original move set from Super Mario 64. It’s for the sake of simplicity. Normal players won’t use most of it in a regular playthrough.
If you want to see Mario used to his full potential, speedrunners embrace his entire arsenal of skills. It can be even more entertaining to them see beat the game without jumping at all.