I know this is an old question but there are a few errors in the answers. To comply with the warranty of the device, tankless heaters do have to be flushed annually with a solution like white vinegar. just like your coffee pot. On natural gas, they would save a lot of power compared to an electric heater. On propane, they probably just eke out an advantage over electric. I would not use a propane standing water heater due to the cost of energy.
Overall, I think the tankless heaters cost more to operate than a standard electric heater but I don't know about the heat pump models.
On demand heaters do have a few other problems. the biggest is cold slices, you turn the hot water on then off then back on you might get a long shot of cold water since the heater has to run water to determine the heating parameters. This will prevent you from doing those on and off shower things but most people don't do that anyway. I think the biggest problem this causes is with dishwashers and washers. Even if you preheat the water by running it at the sink then turn the water off and start the dishwasher on, the cold slice will fill the dishwasher and it will have to reheat the water. In most cases that is not a problem and is probably more efficient than preheating except with some Bosch dishwashers which tend to burn the control board if the internal heater runs to long a cycle.
The second case is a washer that regulates inbound water temperature. The washer turns the hot water on and off and induces the cold slices. I don't know what, if any, impact this has on clothes cleaning.
I have an electric tank and a propane tankless and use them as needed. The tankless delivers hot water as long as you have gas and water and can fill big tubs effortlessly. If there aren't many people in the house, I often use it. But on the whole, we use the electric far more. It is located in a utility room on the lower level walkout basement which usually needs some heat so the lost heat from the unit is not wasted. If I had to choose one, I would take the electric tank if there were no big jetted or soaking tubs to fill (which require two tank heaters).
If I were going to spend a $1000+ on a water heater today, I would check out the heat pump models first. Start at Consumer Reports.