The difference between a moon and a planet is entirely semantic. A planet orbits a star. A moon orbits a planet. That's pretty much it.

The composition of moons and planets has far more to do with their position in the solar system (bodies closer to the sun have more rock and less ice) than with what they orbit.

Consider the planet Mercury. Mercury is smaller than the moons Ganymede (Jupiter) and Titan (Saturn). Nevertheless, Mercury orbits the Sun; Ganymede and Titan orbit planets.

Consider Pluto. Pluto is smaller than no less than seven of the moons in our solar system, including Neptune's Triton, which is thought to be very similar to Pluto and even to have once been a planet.

(It's true that there was recently an abortive move to reclassify Pluto as a Kuiper-belt object -- that is, not-a-planet in the same way that the asteroid Ceres is not considered a planet. But that failed, partly, I suspect, because Clyde Tombaugh is still alive and well-liked.)

For more on the solar system, see The Nine Planets.