As per the province's website
Sublet
When the original tenant allows someone (the sub-tenant) to live in
their rental unit and pay the rent for all or part of the term of the
tenancy agreement.
In this situation, a new tenancy agreement (a sublease), must be
signed by both the original tenant and the sub-tenant. The original
tenant then becomes the sub-tenant’s landlord – both this relationship
and the one with the original landlord involve enforceable rights and
responsibilities. The sub-tenant only has the same rights and
obligations outlined in the original tenancy agreement – the agreement
with the sub-tenant cannot contradict the original tenancy agreement.
Assignment
When the original tenant finds someone to take over the tenancy
agreement – usually to get out of a fixed-term tenancy early or when
transferring ownership of a manufactured home. In these cases, the new
tenant assumes all of the rights and responsibilities under the
original tenancy agreement – unless the landlord and new tenant agree
to new terms or a sign a new agreement.
TL;DR sublet is where original tenant continues to pay the landlord, but the person who now lives in the dwelling pays the original tenant. Original tenant is still responsible to the landlord in subletting. Assigning is where original tenant is completely excused, and the person replacing him takes over all responsibilities and deals directly with the landlord.
I think in BC subleasing is considered the same as subletting.
