No. Both mean the same thing.

yes, basically the same thing. The lease is the rental agreement you sign when you move in. This is where the rental terms are defined. Rent is what you pay each month.

Actually, it depends on what kind of lease you mean. If you are meaning like leasing an apartment vs renting one, yes it's basically the same. But since this is an accounting forum, there are other forms of leases we use in accounting. You can lease other things. Some are treated exactly like rent, where the payment is recorded as an expense. Others are treated as though the item were actually purchased and payments being made on a loan.