Traditionally, the difference between a "partner" and an "employee" is that partners share a formally-defined portion of the earnings of the business. A partnership is a business in which all the profits of the business which aren't simply invested back into the business flow to the partners.

I have no idea if that's exactly what it means at YC; partnership at YC might have as much to do with how portfolio-company-facing or visible a team member is.

"share a formally-defined portion of the earnings"

Which typically vary according to the particular partner by various items including seniority, equity invested, as only two factors.