It depends on the context. For renting or leasing an apartment, there's no big difference. But renting a car and leasing a car are totally different. You rent a car for a few days on vacation. Leasing a car means you have it for 2 or 3 years or more.
In most cases, rent is the better and more useful word. You rent bicycles, rent boats, rent venues for weddings and parties, rent a tuxedo, etc.
Lease always means something long-term and is a more formal word.