In a sublease, the original tenant either moves out of the property and a new tenant moves in for the time period specified in the sublet contract, or the original tenant subleases a portion of the property to another tenant. The original tenant either returns to the property and resumes living there before the master lease ends, or in the case of the original tenant renting a portion of the property to another tenant, the new tenant moves out before the master lease ends.
As an example, let's say you're leaving the country to complete an internship for school and won't return for three months. In order to avoid having to pay rent on an empty apartment for those three months, you could sublease your rental to another tenant. Once you return from your internship, you can resume your lease.
In a Lease Assignment, the original tenant moves out and a new tenant permanently takes over the rental property for the remaining time on the master lease.
For instance, let's say that you decide to move to another city for work purposes, but you have time remaining on your current rental lease. In order to avoid potential penalties for breaking your lease early, you can use an assignment to transfer the remaining time in the master lease to a new tenant.