Assure = give verbal support to a belief: I assure you I'll be there.
Ensure = make sure something happens: He tied her up to ensure that she wouldn't escape.
In the world of insurance (paying regular small amounts of money in return for getting money back if something bad or unexpected happens), there's another distinction to be aware of. You have fire insurance in case a fire happens, but 'life insurance' - insurers insist, although most people use that expression - should really be called assurance, because we are all assured of dying sooner or later.
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