The actual difference
Compare the job each one does.
What problem does it solve?
Building permit: Construction approval and code compliance.
Minor variance: A specific zoning non-compliance that requires relief.
Who reviews it?
Building permit: Municipal building officials.
Minor variance: The municipal Committee of Adjustment or equivalent process.
Order
Building permit: The permit can be issued once required approvals and compliance are in place.
Minor variance: If needed, the variance generally has to be resolved before permit issuance.
Example
Building permit: Structural drawings for a home addition.
Minor variance: An addition proposed closer to a lot line than zoning allows.
A minor variance does not replace the building permit. It resolves a zoning issue so the project can continue through the permit path if the other requirements are met.
Official sources
Ontario — Building permits
Ottawa — Minor variance / permission