Most of the snakebites that I have to deal with here are from copperheads. We do have cotton-mouth water moccasins, but most of my patients are not in the water so much. Usually you just see some facial swelling, as dogs typically go in nose first, and the snake bites them on the lip. You treat them for pain, for inflammation, and give antibiotics, as the puncture wounds from a snake bite can get infected, just a like a dog-bite wound. Puncture wounds are deeper than they are wide, so you can't really wash them. The snake's mouth is dirty, just like your own (well maybe not JUST like, but everybody's mouth is full of germs), so the bite wound can get infected.
Then there is today's case. This is the worst case of post-snake-bite swelling I have ever seen. The owner says that this is actually reduced from the night before when the dog grabbed the snake. This dog usually looks like the Pit Bull on the left here. Now he looks like a Neapolitan Mastiff.