Why is the Moon smooth near the equator and bumpy near the poles? It looks as if it was polished by applying emery to its spinning equator.

My initial guess was that it was simply an illusion caused by light direction, but the terrain map shows the same phenomena.

Google Moon photo and terrain maps (source)



North pole mosaic assembled from photos by NASA's Lunar Reconnaissance Orbiter (source):



South pole shot by NASA's Clementine (source):
There are a few factors that create this effect:

First in the visual light images, the view around the equator is seen back lit, with almost no shadow, the few around the pole is seen with low lighting, and low lighting brings out features of the terrain. The Polar mosaics have an unreal quality, since they are made of images all taken when the sun is locally at its maximum above the horizon, so the light appears to be coming from all directions at once.

The first mosaic and the topographic map have a projection that is "conformal" this means that circular craters appear circular on the map, but it does introduce very large distortions of scale this, combined with shading which has been added to the map, makes regular-sized craters near the poles appear huge. The shading scales with the map, so craters near the pole have much more dramatic shading than those near the equator, that is an artistic effect, not real.

Finally on one face of the moon, that which faces us, and is in the middle of the pictures, there are large "maria". These are relatively flat lava plains. There are no maria at the poles. Most are on the near side of the moon: see Why are most lunar maria on the visible side?

So this is a combination of low angle of light at the poles, the way that google has projected and shaded their map and the distribution of lava plains on the moon. The poles are not more bumpy than the equatorial region in general. They are more bumpy that the lunar maria, but the apparent effect you describe is an illusion.