lease and rent

In what context? Apartments? Cars? Portapotties?

As a general rule, lease refers to a relatively long-term commitment, while a rental is -- again, relatively -- short-term. You rent an apartment and pay a monthly rent on it. But in order to secure that apartment you may have had to sign a one-year or six-month lease. It's the same with anything else. You can rent a car for a few days, but if you wish to keep the same car for a year without buying it, you would have to sign a lease.

There may be other legal ramifications but that's how I understand the difference.

Lease is the contract between the person who owns the property and the person who is paying to use it. The rent is what you pay every month if you have signed a lease with a property owner.