Why you should switch from a conventional water heater that uses a tank to hold and heat water, to a tankless water heater, is really a no brainer. Why in the world would you want to try and keep forty gallons of water hot all day when you might not even be there to use it?
You are then paying for electricity or gas, if you have a gas water heater, unnecessarily. A tankless water heater provides water on demand. Which is another reason to have one. You will never run out of hot water and have to wait till it gets hot again, as those with large families have to do.
Another reason is that the tankless variety takes up a lot less space than one with a tank. Being 2 to 3 times smaller, they can be, and are usually just mounted on the wall.
Additionally, since there is no holding tank, there is very little surface for the accumulation of sediments. This means much lower maintenance and cleaner water.
However, the greatest benefit is that since a tankless heater does not need to be on all the time, you save a lot on your energy bill.