A lease is a written form of rental agreement, which specifies the length of the term, rate, responsibilities of both parties, and other details.
In Canada, these terms are often used interchangeably. Sometimes "Rental" refers to a verbal agreement vs written, and short term vs fixed term, with fewer rights on both sides, but that assumption can be argued.
No, you do not own the house after the lease is up. That would be a Lease-to-Own or Rent-To-Own and is totally different. (Stay very far away from these!! your next question is why -- please ask in it in another thread, there should be a few on here already)

Hope this helps!
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