Your Rental Agreement may contain a clause about assignment and continuing liability. If it does not, the landlord usually decides whether the assignor will be responsible for damages or other breaches of the lease caused by the assignee (or the assignee's guests, clients, or customers).
The Lease Assignment should note whether the assignor is liable for the assignee's conduct (for instance, paying for property damages, missed rent payments, fines from not complying to noise ordinances, etc.). If the assignor has been released from liability, the landlord can only seek compensation for property damage or other lease breaches from the assignee.
In contrast, if the assignor remains liable under the original lease, then the landlord can seek recourse from both the assignee and assignor.
If the assignee is liable but the landlord tries to collect payment from the assignor, the Assignment Agreement will help protect the assignor by stipulating that the assignor can seek recourse from the assignee.