What constitutes a Varsity Sport
D1 & DII give scholarships, and DIII do not; although they can assist in a back-door way as "need based" financial aid or "merit" scholarships.
Basically, when the University does either they typically "officially" recognize the sports team as representing them.
In essence, varsity status is achieved by the backing, and presence of, financial participation with rules and expectations attached.
That said, there are different rules as well regarding who can play, and once a student is entered into a varsity level of participation the NCAA rules will prevail.
Club sports are typically not subject to such rules, and they do not receive athletic or University funding as aforementioned…however, they may, in some cases, receive University funding (through auxiliary budgets like Student Government) as a "club".
The other factor is commitment level…varsity athletes have much more expectation given investment into their direct situation, and experience academic monitoring to remain eligible, while club level has much less or none at all.
On a side note, some small private Colleges and Universities also attach stipulations to even participate in club level sports; as a means to maintain "brand control". Even while not a varsity sport, the club does represent them in name.