While in one way the lease and renting agreements would be almost the same thing they really are different from each other.
When you rent an apartment and sign a lease agreement you're usually leasing the apartment for a long period of time such as 12 months or more and this lease holds you to the property and you can't leave without breaking the lease agreement and facing legal problems.
When you sign a rental contract you are most likely renting from month to month and can leave basically at anytime without any legal problems.