Brushless motors are more efficient because there is no friction loss from brushes rubbing against a commutator, and because electronic commutation (computer control) is more precise than mechanical commutation. The higher efficiency of these motors makes for greater power and longer runtime than a conventional motor of the same size.
Brushless motors run cooler, quieter, and with less vibration. The absence of brush arcing reduces radio frequency and electrical noise that can interfere with electronics. Unlike brushed motors, which are often wired to be more powerful running forward, brushless motors are equally powerful in either direction.
Brushless motors are more durable because there are no brushes to wear out. The lack of electrified windings on the rotor reduces heat at the core of the tool, where it's more difficult for it to dissipate. To prevent the electronic controls from being damaged, some manufacturers encase them in plastic (or "pot" them) to protect them from dust and moisture for the life of the tool.
Brushless motors have a greater power density, so for a given level of output they can be smaller and lighter than conventional motors. Since these motors are electronically controlled, it's easy to add performance features such as multiple speed settings, soft-start, and soft-fade when the tool is switched off.