A written lease, on the other hand, gives a renter the right to occupy a rental unit for a set term -- most often for six months or a year but sometimes longer -- as long as the tenant pays the rent and complies with other lease provisions. The landlord cannot raise the rent or change other terms of the tenancy during the lease, unless the tenant agrees.
Unlike a rental agreement, when a lease expires it does not usually automatically renew itself. A tenant who stays on with the landlord's consent after a lease ends becomes a month-to-month tenant, subject to the rental terms that were in the lease.
Hope this helps.
Web Reference: http://www.nolo.com/legal-encyclopedia
Whereas month to month is where either party can cancel the agreement with a 30 day written notice.
You will most often see the rental start off as a "Lease" and at the end if neither party gives notice, it automatically goes to a month to month.
Rarely will it start month to month unless someone is doing a roommate situation, renting a basement/garage or if someone is just looking for short term as in when a house is for sale.
Hope this helps,
Cassandra Bickel, REALTOR