There's a reasonable argument to be made that, in non-deadlocked congress terms, the First Lady has more power and influence than the Vice President.
There's an old (possibly apocryphal) story about Calvin Coolidge. When he was VP, vice presidents lived in the Willard Hotel if they didn't want to give up their home residences but could not technically live in both the states they're from and the District of Columbia.
So, one day there's a fire at the hotel and the firemen evacuate everyone. Coolidge, in his pajamas and needing a coat, told the fireman he's going back inside. The fireman says no.
Coolidge gets angry, says he's the vice-president, and demands to be let in. The fireman steps aside.
Five feet from the door, the fireman grabs him by the shoulder and says "Wait, what are you the Vice-president of?" Coolidge answers him, the vice-president of the United States.
"Oh," said the fireman. "Then you have to leave. I thought you were the vice-president of the hotel."
VPs get no respect.