The advantage that ESDF movement has (boo, WASD uses the wrong keys), is that you can stop movement by just not pushing any keys or buttons. With mouse-based movement there's rarely a "stop movement" button, you just have to click close to where you already are.

But, the bigger difference is that you can dedicate one hand to movement and the other hand to aiming. This is key when you're playing a ranged attacker. With a melee attacker clicking on an enemy is fine because you want to both attack it and move in that direction. But, if you want to stay away from the enemies and attack them from long range, a mouse-based movement system combined with a mouse-based aiming system means you're clicking on one side of the screen to aim and the opposite side to move. That's inefficient and imprecise.

ESDF movement does have its drawbacks though. The biggest drawback is that you can only move at one speed. That's especially bad for certain scenarios, like escort-type missions where you want to move at the same speed as whatever you're escorting, or stealth games / activities where the slower you move the more stealthy you are. But. for those, the mouse isn't any better. What's better is a joystick / thumbstick. With those you can move the stick very slightly to move very slowly.

Often I think the best of both worlds would be a mouse for aiming and a thumbstick for movement.