Why does boiled water have a different taste as compared to cool water?
The changed taste that stays after cooling the heated water

Hot (boiling) water has a much lower capacity for dissolving gases. When boiling water, $\ce{CO2}$ is removed and through the equilibria behind ($\ce{CO2 + H2O &lt;=&gt; H2CO3}$ and $\ce{H2CO3 &lt;=&gt; H+ + HCO3-}$) so is $\ce{HCO3-}$.

Basically, this removes  $\ce{CO2}$, overall carbonic acid species and also acid equivalents: the resulting water will be more basic (less acidic).

$\ce{CO2}$, $\ce{H2CO3}$ and $\ce{HCO3-}$ are not only present in carbonated (sparkling, fizzy) water but also in tap water. For example, this analysis sheet of the Munich municipal water supply cites 300 mg/l $\ce{HCO3-}$, 15 mg/l $\ce{CO2}$ and 0.5 mg/l $\ce{CO3^2-}$ and a slightly basic pH.

It is known that mammals have $\ce{CO2}$ receptors in their acid taste cells, and this is thought to contribute to the water taste. (I don't have the full text, but Zocchi et al: The cellular mechanism for water detection in the mammalian taste system, Nature Neuroscience volume 20, pages 927–933 (2017) looks promising.) 

BTW, two further candidates for that bland/flat/stale taste are
deionized water and reverse osmosis drinking water.

Warm vs. cold water taste

There will be further changes with temperature, including acid base equilibria and pH, partial pressure of volatile molecules and so on between cold and warm water. And this also applies to our sensing in the taste cells. Put this together with taste/smell sensory input being subject to further processing in our brain to produce the taste/smell impression, and I think it is very hard to compare taste/smell of cool and hot things.