What are the advantages to movespeed and attack speed bonuses being percentages over flat values?

%'s allow items to affect champions differently.

For example, a .625 attack speed on a champion vs a .575 attack speed both buy a recurve bow (+30%), the first gets .1875 bonus attack speed, the second gets .1725 attack speed. This lets it so you can adjust champions to be better or worse with attack speed items or buffs.

My understanding is that the percentage scales off the base attack speed of champions (Not all champions have the same starting attack speed)

So some champions benefit more from attack speed than others.

So you don't understand the difference between a t-shirt being 1.00 off vs 10% off? You should probably discuss this with your elementary school math teacher.