Brushed motors are around 75% efficient, brushless are around 90% efficient. This has two effects: obviously, you get more flight time from your battery with a more efficient motor, but that's not the main effect. The big one is that power output is limited by the combination of how much heat you can remove (which is roughly the same, perhaps 20W) and how efficient your motor is. With a 90% motor, your 20W of heat max means you can have 180W of thrust, while on a 75% motor the same 20W of heat only corresponds to 60W of thrust!

Brushless motors need some high-speed switching electronics to control the motor, swapping from one set of coils to the next at just the right frequency. Brushed motors are stupid easy - just connect two wires, and the voltage corresponds to torque (and thus speed). Any hobbyist could build a brushed controller in their sleep, but it takes a lot of dedication and engineering to design a brushless controller. Typically, this complexity is abstracted away from the end user in a prepackaged Electronic Speed Controller, but most micro quads integrate the ESCs on the control boards.

Brushless motors are more expensive than brushed motors. They're arguably less complex, but we've had many years to figure out how to build super cheap brushed motors, and can make them very small and very cheap.

In general, brushless motors are so much superior that you shouldn't even consider brushed motors in a quad (unless you need to hit a $19.99 price point), but do recall the cube-square law: Length squared is area, length cubed is volume. Area corresponds to propeller swept area or wing area and thus lift, while volume times density is equal to the weight. This is why a hummingbird or sparrow can dart around, while a swan requires a running start and a turkey needs to sprint and buffet (yum) the air with its wings just to get airborne for a few seconds and roost in a tree overnight. And it's why a $19.99 plastic micro quad with brushed motors will fly just fine, but to lift a DSLR you'll need brushless motors and carbon fiber.