The spots come from uneven drying. The only way I know of fixing it are to dampen the suede and let it dry at the same rate. But try at your own risk, as dyed suede will most likely darken/bleed with water.
The step where they take the brush and press it into the towel to get the most water off is key. You want to use as little water as possible, just enough to get a little foam. Then the next crucial step is to wipe the shoes down as quickly as possible with a dry towel with the twist technique.
After that getting the nap back will be the hard part. You can get it looking good but rarely back to the original softness and texture.
You could also find a local shoe repair shop, they should have the necessary tools to clean suede. Good luck!