Marriage Name Change

My wife and I got married 5 years ago. She never changed her last name. We moved to TN recently and we figured we might as well finally change her last name. I have two last names, example Smith-Wesson. We went to the social security office and got her name changed on her SS card. So she is now Mary Smith. We went to the dmv and they said they had to put her last name as smith-wesson. Because thats my last name and that's how my name appears on the marriage certificate. And we said how come the social security office didn't say anything? She proceeds to say "you can put anything on a social security card. You could even put a nickname on there it doesn't matter" But I guess I was wondering who is correct? What can I do?