If itunes finds the artwork for you, it never adds it to your MP3 files, so you have to Get Info and copy and past it. If you have art yourself, then you have to copy and paste them in. A little strange.
n my experience they only way to get the artwork stored into the ID tags is to manually do so.
-If iTunes finds the artwork on it's own, to add it to the ID tag, 'get info' of one of the songs on the album, go to the artwork tab, select the album cover, copy it (command-c in OSX, control-c in Windows), then highlight all the songs on that album, 'get info', click in the blank artwork field, then paste (command-v or control-v) and hit ok. It will then embed the artwork in the ID tag.
-If iTunes doesn't find the artwork and you add it yourself, it's a little easier...just find the relevant artwork (I usually get a 300x300 pixel or larger file; I check Wikipedia first, then I do a Google image search based on album name), copy the picture in the browser (or download the jpeg if you'd like, although it's an unnecessary step) by right clicking (or control clicking if you have a 1 button mouse), select copy image, then 'get info' on all the songs on the album and paste in the blank artwork field like in the above case...which will then embed the artwork you found in the ID tag