Of course, I'm biased because I graduated from ASM (the American School of Madrid, or Escuela Americana de Madrid if you prefer), and received an excellent education there. And, a good friend went to ASP for a number of years.
The difference is that the schools aren't subsidized by taxpayers, but I'd be willing to bet the actual cost per student is about the same; IIRC the typical cost per student for public elementary schools is in the range of $6-9K.
One big reason for going to the American Schools is that they are accredited by American organizations (e.g. Middle States for ASM) and follow American curriculum. This is more important for high school (it makes coming back to an American college a lot easier) or if you're going to be there for a short time, since there won't be rapid changes of curriculum.
But there's still plenty of international exposure, perhaps more since they get students from around the world. The common language at ASM was "Spanglish", and we had students from, among others, Spain, England, Denmark, Japan, Venezuela, Libya, Iran, Israel, and Finland.
My memory is that parents' jobs were a mix of business (including Spanish), embassy (US and others, e.g. Venezuelan), and a little military.
Of course, locals schools are a viable option. For example, I have friends that lived in Germany and sent their children to the German schools -- and they're back in the US and have turned out fine. Based on my (somewhat sketchy) knowledge, European education systems are pretty different, teaching things at different times, etc.
As a side note, I find it interesting that some European co-workers of mine (one German, one Italian) speak excellent English, but don't write it very well at all. I'm not surprised they find English spelling difficult!
Tony