This does vary quite a bit from school to school, as others have said. As far as support from the school, some provide access to some equipment and facilities. Some don't. Some will provide some visibility and organization via school website (not the athletic site). Some don't. The big difference is time commitment. Most club teams don't practice and work out nearly as regularly as D3 (or any other level) teams. There is typically a wider range of player abilities on a club team. Often, they are run by students.
Once in a while, you'll get a few good players who could have played for the school team but didn't for a variety of reasons. But, usually, it is lesser talented/skilled players.