What is the difference between facilitated diffusion with the of a carrier molecule and active transport with the of a transport protein? (osmosis and diffusion)

The most essential difference is that one is active (uses energy) and one is passive (doesn't require energy). take note whenever you see the word 'active' in this context- it denotes that energy is required for it to occur, usually against the concentration gradient. facilitated diffusion is just another passive process by which molecules move from an area of high concentration to low concentration. note that this requires no energy to occur, except it differs from diffusion in that a carrier molecule is required (usually because the molecule is too large to freely cross the plasma membrane)