I think one possible reason why keep a snake as a pet is different from keeping a dog as a pet may be because dogs have been domesticated longer and more extensively than snakes.
Perhaps if snakes were kept as pets for another couple of hundred years, things might be different. I mean keeping the same snake and its offspring for a couple of hundred years, not keeping different snakes for a couple of hundred years. If generation after generation after generation of snakes were bred by man, they may start acquiring characteristics not found in wild snakes. Kind of like dogs acquired characteristics not found in their forebears, the wolves.
Off-hand I can see many advantages of keeping a snake as a pet. The immediate one that comes to mind is the low cost of maintenance. Can you think of any other advantages of keeping a snake as a pet as compared to, say, a dog?