The Moon moves slower across the sky.
If you’re looking down at the Solar System from above, everything turns counter-clockwise, in orbit and in rotation (well, except for Venus and Neptune, but neither of those is relevant for this question). Because the Earth is turning counterclockwise, everything else moves clockwise relative to the Earth. Nothing else of interest has a period as short as a day, so this is the dominant motion in the sky.
But to say which things move faster, we also have to consider the orbital motions. The motion of the Earth around the Sun is fairly slow, once per year, so compared to faster things like the Moon, we can ignore it. But the Moon makes one orbit a month. And this orbit is counterclockwise, making opposite of the clockwise-relative-to-the-Earth motion that everything else shows. Since a one month period is slower than one day, it can’t completely counteract the effect from the Earth’s rotation, but it partially does so, so the net motion across the sky is slower.