Why is the Moon's orbital period different from its phase period?

Good question, its to do with the difference between the sidereal rotational period and the synodic rotational period. Relative to the background stars, the moon takes 27 days 7 hours and 43 minutes to orbit the earth - the sidereal orbit. Ignoring the sun and just looking at the earth from above the north pole, with the moon in orbit around it, this is the time for the moon to circle once around the earth.

During this time though, the earth has progressed around the sun somewhat, on its orbit around the sun. For the moon to get back to exactly the right phase, it takes a little longer, 29 days, 12 hours and 44 minutes in total from full moon to the next full moon. This is known as the synodic period.

This is due to the portion of the orbit, the curve that the earth has progressed it that time. The phase of the moon maybe the same again after this synodic period, but the moon will have made just over one orbit relative to the background stars, and will have moved round slightly to be in front of some different stars to those at the previous full moon.