Tap water is always fresh, and has to pass really stringent safety tests.
Bottled water is often mineral or spring water. For a start it isn't chlorinated, and some will contain bugs - not that this is a bad thing of course, we need natural resistance. Well it used to contain bugs, maybe it gets UV treated nowadays - it isn't an issue though.
Mineral water is from literally the ground, it will contain many minerals, but usually taste.
Bottled water often spends a long long time in storage. That bottled water might very well have sat in it's plastic container for two years - this has been noted. Again, tap is always fresh from the reservoir/treatment works/pipes, call it what you want.
Tap can come from a number of sources. Ours for example comes mostly from boreholes, as we're in the Pennines (A range of hills), we are literally getting mineral water through the tap
There is a nearby mineral water company which will be using and selling the same stuff we get on tap.
Something you will always notice when visiting a different place, is how the water tastes different. You're literally drinking a bit of the local district, as the rocks will have dissolved in the water in minute amounts. Archaeologists can use teeth samples from skeletons to find where that human grew up - the mineral content can be that specific.
So anyway, as long as your tap water doesn't taste too bad, all you have to do is fill a container and put it in the fridge overnight. The chlorine will evaporate, and the resultant cooled water will be indistinguishable from mineral. Well this applies around here anyway, if you have a high mineral content, you may need a filter jug.