The TAS version does a trick in the Deku Tree that results in the one set of bars in the tower escape being up. There's a bitmask in memory that dungeons can use to store flags about whatever they want. One of those flags in the Deku Tree dungeon is whether the ladder in the slingshot room has been shot down. That same flag in the dungeon escape indicates that Zelda has already raised a particular set of bars at one of the tower entrances. Because you warp straight from the tree to the tower, it keeps the flags set, so if the ladder was shot down in the tree the one set of bars is already up in the tower and you don't need to wait for Zelda to open them. This saves some time because she's slow, but in theory you'd still need to run all the way to the door.

Luckily, there's a second glitch that involves falling off the edge of the tower. Normally the game detects when you fall too far and warps you back to the top. If you hit a boulder in mid-air, it resets the fall counter, and you can manage to fall off one bridge and land on the bridge below. The TAS does this twice to fall down to the bridge that has the open gate, so Link can then run right through the entrance without waiting for Zelda (she catches up when you enter a new area). This is the part that Cosmo was describing as impossible for a human, because the boulders are random so you can't really time it such that you hit one in mid-air at the right spot to reset the counter.

The void warp (the new trick used in the WW) is kind of related in that it involves falling off the tower, but that's about it. The entire setup is such that the game detects that you've fallen off the bridge at the exact moment that you hit the loading zone for the inside of the tower. The game tries to reset your position back to the bridge you fell off of, and tries to put you inside the tower, so you end up in the tower but outside the normal rooms, so you just fall all the way to ground level (which is where you wanted to be anyway) and walk to the next loading zone