The finish on a diamond is more of an aesthetic term rather than a term that would change the value of a diamond.
The finish just refers to the polish on the external part of the stone. Most diamonds that you will buy will already have a polished finish to them to give them the sparkle and clean look that you would want in a stone.
Diamonds are essentially created for the commercial market, so you everything from mining to cleaving, to setting it on your ring is done to make the stone more aesthetically beautiful.
This is what polishing achieves when it is well cut and polished, it is said to have a beautiful finish.
Many people think that the finish of a diamond refers to the coating of a diamond. The coating is a treatment done to change the color of a diamond.
A finish is a treatment to smooth down the diamond and make it look better to the naked eye, but will not actually change the appearance of the stone.
It can be done with a rough surface in certain ways to achieve the look that the customer would enjoy. It is simply just a thorough polishing or cleaning. A normal diamond can have a smooth or granular finish.
Neither is better than the other, it is just up to your personal preference. Most diamonds, however, are done in a smooth finish to make it more reflective. Frequent cleanings can help maintain the appearance of the diamond finish.