IO,
What mm_ said (minus some confusion between ellipsoid, geoid, and terrain elevation but it doesn't matter).
Very briefly, a GPS measures time differences from which a series of vectors to known positions in a given reference frame are derived, from which the receiver's position in said reference frame can be obtained and compared to other positions such as that of the ground or an elevation datum. The end result is a geometric distance.
An altimeter is simply a barometer with the scale graduated in distance rather than pressure units. The equivalence between the two unit systems is given by a purely conventional model such as ISA. However, its readout being in feet or metres does not change the fact that an altimeter is solely measuring pressure, or more technically, pressure difference from a calibrated reference.
Therefore, two different instruments measuring two quite different things. Sometimes it just so happens that the numbers might about match.
To the OP:
In VFR flight, the purpose of using altimeter + QNH is to keep you vertically separated from other aircraft (+ airspace), which are meant to be observing the same convention. If you were flying on an area where everyone used GPS altitude as a reference then you would not need an altimeter at all, but I don't believe such a practice exists anywhere. In the meanwhile, if you are flying somewhere where for whatever reason you cannot obtain a QNH/QFE, then you would just simply state your reference pressure (which you will have picked more or less arbitrarily) along with your altimeter reading on any communication, e.g., "X-XXXX at 5000ft on 1020".