Answer
Both facilitated diffusion and active transport are carrier-mediated. Both also transport particles across a selectively permeable membrane.
Facilitated diffusion transports particles down their concentration gradient and does not consume ATP.
Active transport transports particles up their concentration gradient and consumes ATP.
Work Step by Step
Both facilitated diffusion and active transport move particles through a selectively permeable membrane by using transport proteins (carriers).
Since facilitated diffusion is down the concentration gradient (similar to regular diffusion), no ATP is needed.
Active transport, on the other hand, transports particles up their concentration gradient. Anytime something is going up its concentration gradient, energy/ATP is required.