What's the difference between "any" and "every"?

"Every" is closer to "all". "Any" does not imply "all", and perhaps only one. "Every" has a more individual connotation (such as individual attention) than "all" (more like a blanket condition).

Any chooses one item from a set (or a few)

Every chooses all items from a set.

Consider everywhere vs anywhere, the set is places (from the 'where'). Everywhere means all places in the set. For example "I dropped the brush in the paint bucket and the paint got everywhere" means that the paint is now in all the places or in every place or everywhere. Whereas anywhere means just one place in the set, but we don't know which place for example "I have forgotten where I left the paint bucket, it could be anywhere by now." The paint bucket is only in one place, but we don't know which place, therefore it is in any possible place which means it is anywhere.