Interesting. If you look at Wiktionary or if you prefer, your favorite etymology dictionary, the word utilize is descended from Latin from the French word utiliser, via the Italian utilizzare which got it from the Latin utilis. All of those words mean to use.
I'd be the last person to say you're wrong. Matters of grammar and usage ultimately boil down to does it feel right and current usage. As is usual with these things, other people have different feelings about it. That's what dialect is I think.
No part of your example indicated the intent was to use the computers to teach children. Teachers can teach adults and they can use computers to do other things. In your examples, excepting your hidden intent, they were synonyms. Do you have a better example demonstrating this?
Nevertheless, "use" is a better word. Using longer words, when shorter words are available that mean the same thing, comes across as pompous or pretentious.
They're not quite the same thing. One example I see online is using "utilize" to suggest that something is used beyond its intended purpose. (I'm not sure I even completely buy that.) But, in general, "use" is shorter and sounds less jargony.
I'd be the last person to say you're wrong. Matters of grammar and usage ultimately boil down to does it feel right and current usage. As is usual with these things, other people have different feelings about it. That's what dialect is I think.