I lived in San Antonio for five years.
I could make the argument in both directions.
Personally, I think San Antonio has better food. Tacos everywhere, the best coffee in the world (according to Alton Brown) and some great unexpected hole-in-the-wall Korean joints. Plus, the new generation of higher-end restaurants like Barbaro and The Granary. Austin will have more variety but I find many well-regarded Austin restaurants to be very overrated (though some merit their fame).
Austin certainly has better night life, though SA's scene is growing (Paper Tiger is a important recent addition). The St. Mary's Strip is closest to Austin's East 6th, but much smaller. Southtown is nicer (it's the hipster area). If you want a show, sit down at Lulu's at the south end of St. Mary's around 1am, order a chicken-fried steak the size of a history textbook and enjoy the people watching as the clubs close down and the show comes in for a bite.
I also think SA is the more beautiful city, especially the Missions south of town. Rent a bike and you can follow the whole trail along the river.
The key in visiting San Antonio is to never set foot on the Riverwalk. The food is awful and everyone's drunk. Frankly, I wouldn't even stay downtown. Splurge on the new Hotel Emma at the Pearl.