pH Difference Our Taste Buds Can Recognise
First, the pH of water changes with rising temperature because the total number of $\ce{H+}$ ions (well, solvated hydronium ions, really) increases, not because it has become more alkaline. The water is still neutral, having an equal number of hydroxyl ions as hydronium ions. Heat simply increases the ionization.

Second, it depends on the individual taster, but a sample of water at $\mathbf{20~^\circ C}$ with a $\mathrm{pH}$ of $6.63$ (perhaps acidified with citric acid) would probably taste sour in comparison with a sample of water at $\mathbf{20~^\circ C}$ with a $\mathrm{pH}$ of $7.47$ (perhaps with a bit of baking soda, $\ce{NaHCO3}$).

See Temperature Dependent[sic] of the pH of pure Water for a more detailed explanation.