Ive been teaching some underclassmen how to play smash, and every single one of them has asked what the usefulness of wave dashing is. I thought of some reasons.
1. fluidity and more control over movement. The wave dash adds another dimension to the movement possible in this game, by adding instant direction and velocity changes to the dash and walk. Instead of dash dancing the whole time, the movement appears to be more random increasing its potential.
2. The direction change can be used to fake out opponents. Wave dashing backwards in an approach is an easy bait to open up the opponent. One can also stand in front of a person getting up, wave dash backward to avoid the get up attack, and then rush forward to attack.
3. It allows you to do any aerial, smash or tilt much faster. If a player waits till a regular dash stops to do anything, there will be more lag, which means more opportunity to be punished. A wave dash can be performed at anytime during a dash, and wave dashing has much less lag than the end of a regular dash. Most moves can then be performed very quickly after the wavedash.
Those are a few reasons that I have come up with.