The Moon is in synchronous rotation, meaning the same side of the Moon is always pointing towards the Earth. My intuition is that it wouldn't make any difference (other than the fact that the whole surface of the Moon would be visible). But maybe there is something I didn't think of.
What would be different on Earth if the rotation of the Moon wasn't synchronous?
It would be noticeable from an astronomy perspective in that we would be able to see all faces of the Moon.
However, it would have a negligible effect on our tides. The Moon is not a gravitationally symmetric body (see mascons), but our Earth tidal deviations are mostly due to the Moon's eccentric orbit and the positive or negative interference with tides due to the Sun.
On the Moon, significant tidal deformation would be taking place due to the Earth's gravity. Some of this energy would be released as heat, so we might also get to see volcanic activity on the Moon.
Over a longer time period, as the Moon tidally locked, we would expect to see an evolution of its orbit, either inward or outward, depending on its spin direction.
