Another way to view the difference between lease and rent is that in the case of rent, there is no defined term. Thus the tenant might say they plan to stay for a year, but they can leave early without any penalty except sometimes a landlord will ask, in advance, for "First and last month's rent" in advance which doubles as a security deposit for damaged property. The landlord can also evict the tenant at any time, since there is no lease arrangement. In practice, in the U.S., the landlord has to give some notice about eviction which varies according to local laws.
A lease on the other hand, puts the tenant in the position of being obliged to pay for the whole term of the lease, even if they move out, unless there are provisions in the document allowing the tenant to leave early with some specified penalty or cancellation fee.