Instant hot water - well yes, sort of. If the heater is a long way from the point of use you will experience the same delay for the hot water to reach the faucet as you have with a tank heater. In fact it may take a little longer as it takes the heater a few moments to get up to speed when it’s called upon. It may be that a tankless unit can be located closer to the point of use than a tank heater because of its small footprint and its ability to be installed on an exterior wall, in which case you may be able to get a quicker delivery at the faucet.
Endless hot water - yes, you won’t run out as long as the fuel supply holds good. Handy for when you have house guests, can be expensive if you have teenagers addicted to endless hot showers.
No standby losses - this refers to the heat lost and energy wasted by a tank heater during the relatively long periods when it’s standing idle waiting to be used and all that stored heat is slowly leaking out to the environment. This is much less of a problem than it used to be as modern tank heaters are now far better insulated than they were in the past. Standby losses are still a factor but a minor one, so don’t count on the energy savings for any appreciable payback on the substantial extra cost of a tankless system.
Economical to run - here we get into some complicated territory. Natural gas is currently a relatively inexpensive fuel and if it’s available at your home it will in many cases be the most economical fuel for both your home heating and hot water needs. This will be the case whether you have a storage tank or a tankless heater. Gas-fired tankless heaters typically run at a higher efficiency than the storage tank equivalent, and this will result in some savings, but if you live in a hard water area that saving is offset by the need for regular descaling of the tankless unit.
If piped natural gas is not available the choices for a tankless heater are not so good. Tankless heaters may be run on propane but this is not such a cheap fuel, moreover it's a petroleum byproduct subject to the fluctuations of the international market, so your propane costs might go through the roof at exactly the same time as you are spending more for the gasoline in your automobile. Not so good.
So what about electricity as primary fuel for a tankless heater? Here the problem is the enormously high current draw of an instantaneous heater capable of supplying an entire home. Regular storage tank heaters replenish the hot water over time and can do so with relatively low power elements with a circuit demand in the region of 30A, while tankless electric heater needs that energy all at once with a current demand as much as 150A. Few houses have that to spare, so you will have to figure an upgraded supply to your home as part of the installation costs.
Environmental benefits - are they green? Here I’d have to say a qualified no. If running on natural gas the environmental impact of tankless heaters is pretty comparable to any other gas-fired appliance. Yes, we're talking fracking and long-distance pipelines to bring that natural gas to your home. If electricity is the primary fuel a storage tank water heater is the hands-down winner. First of all, a tank water heater is an ideal complement to a PV solar installation, serving as low-cost energy storage. Water heated during the daytime and stored in a properly sized well-insulated tank is there ready and waiting for all the morning showers. Furthermore, as the electricity grid moves toward demand management to allow the growth of both distributed and utility-scale renewables the non-negotiable surge demand of electric tankless heaters is highly problematic.
Bottom line: if you want the luxury of limitless hot water and have a natural gas line we won’t discourage you, other than to say that the tankless premium might be spent on more productive ways to enhance the comfort and environmental performance of your home. In all other cases we strongly recommend a high performance storage tank heater, and make it an electric one if you're thinking of going solar.