In Free Play, you and a friend (or a CPU) will run through a contraption in which you keep its balance correct on both sides so that neither of you fall. There will be a checkpoint halfway. When you reach the end, the Miis will press two buttons in the shape of gems on the floor by stepping on them. They will glow when stepped on and will slide down slowly. That will open the door and the Miis will then run out of the temple.
In Challenge, you must go through 10 stages of keeping balance to a platform with another human player, with it getting harder to do so the farther you go. When you make it to the end of every stage, 30+ seconds will be added to the amount of time you have to complete the other stages. Stage 5 is where it starts getting a little tricky.
There are no checkpoints, unlike the freeplay version, so if you mess up, it's back to the start of the stage with you.