Does filtered water really change how coffee tastes?

Yes, though the size of the difference depends on what is in your tap water to begin with.

The change you will definitely notice

Chlorine. Municipal supplies are treated with chlorine or chloramine, and both are perceptible in coffee at concentrations well below what you would notice in a glass of water. They flatten aromatics and add a faint swimming-pool edge.

Any carbon filter removes them almost entirely, and the difference in a light-roast filter coffee is obvious side by side.

The change you may notice

Hardness. Very hard water dulls acidity and mutes the brighter notes; the cup reads as heavy and flat. Reducing hardness brings back definition.

But there is a limit in the other direction, and it matters: strip too much and extraction suffers, because magnesium and calcium actively help pull flavor into solution. Water that is too soft makes hollow coffee.

The change that matters most

Neither of the above. The machine. Filtering roughly doubles the working life of a coffee machine on hard water for around thirty dollars a year — a return no other purchase in home coffee comes close to.

Where to notice it

Filter coffee, light roasts and black coffee generally, because nothing is competing for attention. In an espresso with 150 ml of milk in it, the water difference is close to invisible.

Which filter

A pitcher filter is cheap and universal and requires you to actually use it. An in-tank cartridge is more convenient and locks you into manufacturer prices. A mineralizing filter removes hardness and adds a little magnesium back, which is the version aimed at the cup rather than the boiler.