If you care about the school, go with Stanford. It no doubt has more prestige nationally and internationally. But UW is an excellent school, too.
I'm a local San Franciscan and have lived in the South Bay as well. I can tell you that Palo Alto is an expensive place to live. Going to Stanford for only 5K/year is great but you also have to consider cost of living. Also, you will need a car. The only place in the North Bay where you can survive without a car is San Francisco. Even so, I still have a car and motorcycle for convenience. Outside San Francisco, such as Palo Alto, you cannot rely on public transportation... they come in half to one hour, if you are lucky. Remember, this is California. The commute along freeway 101 is almost always bad. Yes, it is near San Francisco, but it is still 20-25 miles on 101. And once you hit the city, you will meet traffic jam again before you can find parking (which is another adventure). So, it is not a 20 minute ride. Your best bet is to live on campus or near campus so that you don't have to hit the freeway, at least not too much, to go to school.
Seattle is not that bad when it comes to commute. Since you are local, you would not have logistics issues.
I have been to Stanford and UW campuses. So I know both. Stanford is huuugge. You cannot simply walk from one campus to another whereas you can do this at UW (UW is big, too, but still manageable on foot).
Both campuses are beautiful although Stanford is prettier.