I recently commented on this ID problem on MassBird, pointing out differences in details around the eyes.
Chipping sparrow has very distinct, narrow whitish arcs below and ABOVE the eye, contrasting with darker gray-brown feathers, and broken at front and BACK by the dark eyeline. Simply looking for these distinct white arcs is a good quick ID clue, not shared by any other sparrow. (But don’t go running to your Sibley Guide to look it up, I didn’t appreciate how useful it was until recently, so it’s not illustrated very well in the current edition)
On Clay-colored there are pale feathers all the way around the eye, and these blend into just slightly darker feathers, not contrasting at all above the eye (just a broad pale eyebrow stripe). And there is only a very weak dark eyeline breaking the eyering behind the eye.
This leads to the ‘open-faced’ impression on Clay-colored, since the eye is set in a broad pale area.
After reading my comments, Phil Brown put together a nice comparison of two photos, which you can see on his blog here: http://birdsofessex.blogspot.com/2011/10/sparrow-identification-chipping-clay.html