Not in real estate. There is technically no difference between leasing and renting, other than the things that are colloquially associated with. A lease agreement (or rental agreement) may or may not have a clause offering the borrower an opportunity to take ownership.

Rent might also refer to the price portion of a leasing agreement (or rental agreement). So in shorthand usage, “the lease” might encompass all terms of borrowing something, including the price, but “the rent” might just refer to the price.