The decision to purchase, make payments toward, or rent a water heater all depends on how much money you're prepared to hand over immediately and down the road. Some options come with long-term security and easy access to service, while others mean ownership and freedom from contracts.
Rental water heaters
A company owns the physical system, usually a storage water heaters, and charges you a monthly rental fee. You'll also pay the regular charge for fuel usage. This method is generally an option with gas companies. Rental is great for people who don't want the hassle of maintaining their water heater and paying out in the event it needs repair or who don't have the money to purchase a system outright.
As with rental units, leasing usually means more accessible service and less money upfront. On the downside with leasing, you're in for a long-term agreement and you could be looking at an expensive buy-out in the event you decide to switch energy suppliers.
Positive side of renting / leasing
Own your own
If you've got the money to buy a water heater upfront, you'll never have to worry about paying extra each month to rent the system. You will also likely have a lot more choice in water heater types and can purchase the system from a retail outlet, dealer, distributor or wholesaler. Options including solar and heat pump aren't likely to be available as a rental unit. If the water heater (depending on the type, of course) lasts to the high end of the expected range (10 to 20 years), you'll have a good chance of seeing savings. But if problems arise that aren't covered by warranty, you'll be paying out of pocket. And you'll be on your own in finding a service provider.