When a tenant assigns a lease, they permanently give up the benefits of the lease, and a new tenant assumes the lease interest. The original tenant still remains liable, behind the new tenant, unless they are released of this obligation by the landlord.
Subletting a space means to transfer lease rights temporarily until the remaining lease term is over or until the original tenant returns.
With a sublease, the original tenant is still responsible for the lease term, paying rent to the landlord, and dealing with repairs or other issues in the property, whereas a Lease Assignment puts the third party in direct contact with the landlord to pay rent and assume a business relationship.