Subletting and assignment of a tenancy are terms that are often confused. Here are some of the ins and outs for both of them. Sublets are much more prevalent in Ontario, so I will mostly focus on those.
Subletting
Under a sublet, the tenant allows another person to take over the use of the unit on a temporary basis. The original tenant plans to return. Consent of the landlord has to be sought. Situations with multiple tenants or roommates are generally not considered sublets. When submitting, the original tenant continues to be liable for the rental unit and remains the main contact for the landlord. The landlord continues to collect the rent from the original tenant. Taking rent from the sub tenant could be seen as establishing a direct tenancy between the landlord and the sub tenant, opening the door to a plethora of problems.
While the tenant must obtain the landlords consent to the sublet, the landlord cannot arbitrarily withhold their consent. Once consent is obtained, the subtenant has the right to occupy the unit up until the specified end date, giving the original tenant the right to take back occupancy on that date. Since the original tenancy agreement will stay in effect, any possible notices from the landlord will have to go to the original tenant (not the subtenant). It is advisable that the landlord make sure the tenant and subtenant sign a written subtenancy agreement with a clear termination date (move back date). Best practice would be for the landlord to refuse the subtenancy unless such an agreement is in place.
For further info see Section 97 of the RTA.
Rental Assignments
An assignment means that all rights and responsibilities of the tenancy are being transferred to a new tenant. In other words, the whole agreement is transferred to a different person and the original tenant gives up their rights and responsibilities regarding the rental unit. The landlord has seven days to respond to such a request by the tenant. You can refuse the assignment as such or you can refuse the potential new tenant, if you deem them unsuited.
Consult Section 95 RTA for more details.