This article in the Journal of Geodesy explains that it is caused by the local deflection of the vertical in London, due to regional variation in the thickness and density of the Earth's crust.
What Airy's Transit Circle in the observatory in Greenwich defines is not directly a place on the surface of the Earth, but a direction in space, namely that of the local zenith, or plumb line.
The article above argues that this direction is actually, with reasonably good accuracy, still the direction of the modern IERS zero meridian used to define WGS84 and thus GPS coordinates. The various changes of definition over the last 150 years have kept good continuity with it.
However, even though WGS84 uses this direction as a reference, the x-z plane in WGS84 geocentric coordinate is parallel translated 102 meters such that it passes through the Earth's center of mass.
So the real difference is not merely about using a different origin for your coordinate system, but using a different principle for what the coordinates mean. In Airy's time there was no way of measuring the location of the Earth's center with much precision -- what one did was measure the direction of the local zenith and declare that direction (relative to the pole and to the orientation of the transit circle in Greenwich) to be one's coordinates. On the other hand GPS measures your position directly, and the center of the Earth relative to the satellite orbits is well known. In order to express this position as old-style coordinates, the GPS unit will compute a "virtual zenith line" as the normal to the reference ellipsoid that passes through you. The difference between this normal and the IERS zero meridian (which, again, agrees with the orientation of the Airy transit circle) becomes your coordinates.
So when you walk out into the park in Greenwich to find the place where your GPS receiver shows 0 degrees longitude, what you've actually found is the point (modulo latitude) where the normal to the WGS84 reference ellipsoid is parallel to a plumb line erected at the observatory itself.
Neither continental drift nor random accumulated errors from definition changes have had effects comparable in size to this.