A lease assignment is where a new tenant takes over a lease, while subleasing is where an original tenant creates a new lease with a subtenant.
With a sublet agreement, the landlord may still consider the original tenant liable for upholding the terms of the lease. For example, if a subtenant misses one month's rent, the original tenant is still responsible for paying the landlord on time.
With an assignment, the landlord can choose to release the original tenant from all liabilities. So, if the new tenant misses a rent payment, the landlord would not seek payment from the original tenant but from the new tenant.