Here's my understanding (I'm sure someone will correct me if I am mistaken, but I don't think I am. I welcome any criticisms though.)
To Lease means that you have a longer fixed term arrangement for possession of a property.
To Rent basically means that you have either a month to month or just an arrangement that can be terminated at any time by either party (each according to the laws of their state).
Think of the difference between renting a car or leasing a car and it'll come to you.
The other thing is just a phone number.