One side of the coin, A tankless water heater saves energy by only heating water that you are using at that time without any storage of unused energy. If you do not demand hot water on a regular basis or for a continued amount of time, you can save some money on energy bills by not wasting energy to keep water warm until you use it. They take up no floor space and are very neat and tidy mounted on a wall.
The other side of the coin, A tankless water heater has to see a call for water before it will start to produce hot water. If you take prolonged showers the unit will run the entire time to keep up with demand, if more than one shower is used at a time (or a performance shower) volume may overwelm its ability to satisfy both locations. The startup pause (waiting for water to arrive) is slightly longer than a storage tank heater, but probably not noticable to most. May not be DIY friendly depending on owners ability to troubleshoot.
Writers conclusion, These units fall under the same catagory as the flourescent light bulbs that are touted as green. The application needs to fit the use. Flourescent bulbs only consume less energy if they are in locations that a light will remain on for a long enough period of time to offset the greater amount of energy it takes to start one up. Tankless water heaters work on the opposite thinking, you will save more money if you do not have to use it all the time.
When customers of ours are looking into the possibility of adding one of these units we will sit down and present the above information and then ask then what there daily hot water usage is, this is not an exact number thing, just how many showers a day, dishwasher usage per week, washing machine usage per week and then we add up all these items to get an idea of how long this unit will run on an average day. Compare the Btu's of the tankless against the storage heater to get an idea of consumption for each unit, and that is your answer of if it works for you. Your local plumbing supply company have people on staff that can do these calculations for you if you provide the water usage in GPM of each faucet and water consuming appliance as well as duration used along with frequency. They typically sell both so there is no pressure to sell you on one vs the other.
We have installed many of these units in our customers homes that are second and third homes that do not get rented at all and see use only a few days out of the year. We do not put many in primary residences due to the amount of water used on a daily basis.
I hope that helps and was understandable.
One more thing, you may want to compare an electric water heater (being that is what you have now) against a gas unit first to get the better of the two before moving on to compare with the tankless.