Washington University is ranked highly in some undergraduate rankings because by many measures, it provides among the top undergrad experiences in the United States – smart students, small classes, great food and housing, good access to professors, relatively fewer TAs teaching, high grad rates, etc.
Meanwhile, the University of Washington (and Wisconsin, similarly) is ranked highly in rankings that reward things like scholarly awards, publications, and research expenditures/impact: faculty- and research-based rankings, and closely related “prestige” rankings. Schools like the big UWs have cultivated national and international prestige, over many decades, by making scientific breakthroughs, innovating new technologies… basically making the world a better place, and having faculty and researchers recognized for it with awards.
You can get a great education at either and have a blast at either, but they are obviously valued for different things.
I would have a hard time turning down a full ride to one of the top 15-20 private universities in the US. As mentioned, you’d have smaller (and more small) classes, greater access to professors, and likely better food and dorm rooms. Those are some of the typical advantages that the top-notch private schools have over the state schools. But that’s me, and this is obviously your decision. So:
Both schools have plenty of quality programs: if you changed your mind, you wouldn’t be stuck.
St. Louis vs. Seattle is in the eye of the beholder. Certainly Seattle is more picturesque overall than St. Louis, but there’s plenty to do in both cities and the Mississippi River has its own charm. Seattle has rain; St. Louis has heat.
The U of Washington campus is much larger with quite a few more students and more clubs than you can probably count, but WUSTL is not so small that you couldn’t join different types of clubs, have a diverse group of friends, find plenty of things to do, etc.
I’d choose based on fit: academic fit, city/culture fit, social fit, weather fit, and financial fit.
Visit both and try to use that experience, plus your own “fit” checklist, to decide.
They’re both good but they are quite different. Figure out which side of the equation you prefer and don’t look back.