First of all, your question is fair, and since I have kids of my own, I do not take offense to someone trying to figure out what is best for their family.
That being noted, Austin is one of the most diverse populations in Texas (and the South for that matter), so there is no need to worry about 'yankee-hating.' While I am a native Texan who has lived in Austin for 15 years, my mom is from the midwest, and most of my good friends are transplants from D.C., Minnesota, Wisconsin, Philly, etc. So, you won't be the first or the last 'snow bird' to make a home down south.
Sure, there may be the occasional close-minded jerk who feels obligated to share his opinion, but I am sure that if I moved to Ohio there would be some fool who would find it clever to call me a 'hick' or 'bubba.' For the record, Austin is a highly-educated city, so your chances of running into one of those guys are few and far between. Additionally, with your military background, you have served your country and already earned a lot of respect no matter where you come from. You would be welcome here and would experience a crap-free existence.
If you want my honest opinion, I would recomend choosing Austin over San Antonio. As I mentioned before, Austin is more diverse, but it also offers a more laid back attitude, good schools (with an awesome transfer policy), great parks and swimming holes (very outdoorsy and active), fun festivals, great restaurants, good museums and art exhibits, and a variety of nightlife options for when you and the wife need a break from the kids. Plus, San Antonio is only 60 miles south of Austin, so it makes for a nice weekend get-a-way every now and then.
I know that someone mentioned the cost of living (San Antonio vs. Austin), and I am not sure where they got their information. The housing market in Austin is great right now especially in South and Southwest Austin (which is the cooler/more traditional part of town anyway). Sure, there are some expensive neighborhoods in Central and West Austin, but that would be no different than the pricey neighborhoods of San Antonio (Alamo Heights and others).
I hope this helps, and good luck to you and your family on your potential move!