Are ice diving classes more or less thorough than intro to cave? How does the money part play a role?
Way less thorough for diving skills than intro to cave. It's all about how to cut the hole, how to mark the hole, how to tie yourself to your instructor, line tending, and don't pre-breath the reg on the surface. A rock could take an ice diving course and pass the diving portion. A typical ice diving course sometimes involves a couple hours of class room (sometimes not) and 4 shallow and short (15-20 minute) dives.
Money enters for 2 reasons- It can be a good revenue stream for OW instructors in the winter time when open water is not available.
It can be taught without taking a course. Most of the OW guys that I know become ice diving instructors by helping out on a couple of courses then telling their agency that they want to be ice instructors. Much quicker than going the intro to cave instructor route.