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Alex, SE Editor-in-Chief here.
The "advanced" epubs are epub3 files that basically assume your ereader is as fully-featured as a modern web browser. They use advanced CSS selectors, SVGs where possible, and make other assumptions for an "ideal" ereader that are basically not true for most/all ereaders on the market today. They're just the ebook's zipped up Git repo source without further modification. If you visit an ebook's SE web page you'll see an option to read the book in your browser, and that file is also the (mostly) raw ebook source as present in the Git repos.
The "compatible" epub takes the "advanced" epubs as the base, then applies various compatibility affordances to make them actually usable on most of today's ereading devices. Selectors are simplified into classes, MathML and SVGs are converted to PNGs, additional compatibility CSS is inserted, and so on. Unlike the "advanced" epubs, the "compatible" epubs should look good in most modern ereaders.
AFAIK the only ereader to do a passable job on the advanced epub is iBooks, and even then it's not perfect. Desktop web browsers render them perfectly but of course a web browser isn't really an "ereader" per se.
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