I recently installed a whole-house tankless HWH. So far, I like it a lot.
Some have mentioned the small per-faucet heaters. However, the ones mostly promoted now are central tankless. (So your question "are tankless cheaper than central" does not make sense. Most tankless HWHs are central, although as with tank HWHs, you can install multiples for convenience.)
I may save money, but I do not expect a useful ROI. I did it for convenience and to regain floor space. In my 1953 house, the original HWH was in the garage. When the garage was converted (a religion in this neighborhood), the HWH was moved to a utility room even farther away. It was then about 50' from the kitchen and bathroom, and it took nearly a minute to get hot water to either place. It was replaced once after that -- I think in 1991, but it's gone now and I didn't save the info. I suspect that in the original house it was gas, but when I bought the house it was electric. I had been keeping the temperature in the HWH just high enough to shower with hot only, and turning off the lower element entirely, so I had already minimized the energy cost of hot water. Plus it's a one-or-two person situation (two when I started thinking about it, one now), thus correspondingly less savings.
But the part of the house near the kitchen and bathroom simply didn't have any place for a WH. The crawl space and attic are too small -- a HWH on its side might have worked, but I'm not aware of any made to work that way. However, mounting a tankless unit on the wall just outside those rooms was trivial. And the cold, hot, and gas pipes were already in the crawl space (which is a generous, sit-up-in space).
So by switching to tankless, I would get much faster HW, and regain about ten square feet of floor space.
In late August, I learned that my city was currently doubling its usually $675 rebate for switching from electric to gas HW. The double rebate fund was nearly empty, so I had to move fast. In any case, the need for water (under pressure) and gas plumbing put it way outside DIY for me, so I located a contractor experienced in the installation who could do it quickly. He pulled the permit on the last day for the double rebates -- the permit date, not the completion date, determined eligibility. His bid was $2308, and I eventually got a $1350 rebate for the electric-to-gas switch (not related to going tankless).
Since I was in a hurry, I went with what the contractor had (Rinnai), rather than investigating extensively. Other brands I read about were Paloma (aka WaiWela?), Rheem, Tagaki, and Bosch (Aquastar).
The whole thing went smoothly for something distinctly non-trivial. Remember that I was moving the heater (so modifying cold and hot plumbing), tapping into gas plumbing, adding an electrical circuit (for the controller and igniter), removing the hold HWH, and capping the pipes that connected to the old HWH. The main electric panel is the original, and the electrician looked at it and said "I don't even know what kind it is" ... luckily it's a subpanel to a newer panel outside, where it was much easier to add the needed circuit. The HWH did not require a vent -- I'm not sure whether the inside models have to be vented (probably so). I gave the old HWH to a friend who was refurbishing a house for a low-income person.
I got a Rinnai V53e, the smallest in the Rinnai line:
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It is rated for 0.6 to 5.3 GPM at 35F rise, giving a claimed capacity of two simultaneous showers. With only one shower in my house, this was plenty. The low end turns out to be more of a limitation; I'd often like to have hot water at a lower flow.
Having the hot water flow so soon was great. However, I found it difficult to regulate what I wanted. I initially did not have the remote control, which had two disadvantages. First, I didn't know whether I was drawing water fast enough to activate the HWH except by waiting to see if I got hot water. Second, since the default temperature setting is 120F (and the only alternative 140F), I had to mix hot and cold for a shower. This meant I had to run the minimum amount of hot water PLUS some cold, and I normally don't run that much water for a shower. If I ran just enough hot to activate the HWH and then added cold, the back pressure from the cold would sometimes be enough to turn off the HWH. Changes in cold water temp still affected my shower temp. Overall it was a bit frustrating.
I found a remote control for half price and installed it myself. It's trivial -- the most complicated part was drilling a hole through my brick wall. Attached two wires to the HWH and the remote just started working, no setup needed. I love it. It's right next to the shower, so I set it for the temperature I want, turn on the hot water until the light shows me the HWH is active, and enjoy. Never turn on the cold water faucet at all. Output temp is very consistent. (Input temp surely varies depending on whether the water has been in pipes under the house, underground in the service lines, or in the water mains -- though it doesn't vary nearly as much in north Florida as it would a few hundred miles north of here.)
I can change the temp while I'm showering, in 2F increments from 90F to 110F. I have already found temps from 102F to 110F useful. In the summer I will probably go a little lower, but not much. Above 110F, increments are 5F, and you cannot move it above 110F while the water is running -- safety. I have showered at 115F -- that's very hot but sometimes desirable -- but to get there I have to turn the water off briefly to set that temp.
Now I want a remote control in the kitchen too. BTW, while the water is running, one remote has priority and others cannot change the temp setting. Changing priority requires stopping the water flow.
As I said, I did it for convenience and floor space rather than energy cost. I probably will spend less, but in other locations, you have to check the relative cost of electric vs gas energy. I'll also spend less because the new location means I'll waste less hot water left in the pipes. But I may use hot water more often in the kitchen and bathroom, which would tend to cost less. I'm not losing sleep over it either way. At times I may be able to set the temp to 140F to get a pot of hot water for beverages -- will try that in a few days.
Cons? Sure. Certainly more expensive than a tank HWH, though don't underestimate the cost of a gas tank HWH. And remember that my cost included a good bit of plumbing work, not just installing the HWH. I suspect it's not that much more than a comparable tank installation.
When electric power goes out, I lose HW immediately -- I have about eight seconds worth in the pipes. This is unlike a gas tank HWH, which will operate indefinitely without electric power, and unlike an electric tank HWH, which will usually have a tank full of HW when the electricity fails. One of the Bosch/Aquastar HWHs (1600H) has a piezo-electric ignition powered by the water flow, and operates with no other electric power. I like the idea but that unit has no remote control and little temp control.
In a cold climate, you have to be careful that an outdoor unit doesn't freeze. It has built-in protection, but there are caveats in the manual which I skimmed over quickly since I don't have to worry about them.
There are times when I want less than half a gallon a minute of hot water, even with the temp set so I can use pure hot. I may want a slow stream at the kitchen sink. I may even want a very slow shower. Generally, though, these are infeasible with my previous setup, and possibly only a circulating HW system would satisfy these desires.
The rated flow rates are for a 35F temperature rise. Since I never need anywhere near 5 GPM and my cold water ... well, I'm guessing, but I suspect it seldom drops below 60F ... isn't very cold, so I'll never run out of 110F water. A household up north, with probably 40F or lower winter cold water temp and needing two simultaneous showers, might hit the limit. OTOH, such a household might have trouble keeping enough HW from a tank HWH.
Really needs the remote control in each use location for greatest benefit. This of course increases the cost, and in some households would probably bring up user conflicts. (OTOH, if simultaneous users are that common, perhaps it's better to have only one control and not to count on having fine temp control, as I do.)
If you already have gas, you probably have an adequate supply. If you don't have gas, then you have to add it or else have a large electric feed (as others have described).
Note that on my unit, the trip-on time is only half a second, thus very little water lost to the trip time. But if you're trying to get just enough to trip it, you are likely to waste some water while ramping up the flow.
Edward