TL;DR: For all practical purposes except milliarcsecond radio astronomy, TAI and GPS time are separated by exactly 19 seconds.
TAI and GPS time both attempt to represent time as ticked by an ideal clock at mean sea level on the surface of the Earth. Both differ from UT1, time as ticked by the Earth's rotation (this is the source of leap seconds), and because of how they represent time, they also differ very slightly from one another.
GPS time is an operational time scale. Each GPS satellite contains an atomic clock, but with a tick rate that deviates slightly from the standard definition of "9,192,631,770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the caesium-133 atom".
The atomic clocks on GPS satellites are subject to both special relativistic and gravitational effects. The special relativistic effects result from the relative motion of the GPS satellites and a clock on the surface of the Earth. The gravitational (general relativistic) effects result from the altitude at which GPS satellites orbit. To compensate for these effects, the GPS satellites' atomic clocks are intentionally made to tick at a slightly slower rate than is standard (the standard being "9,192,631,770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the caesium-133 atom").
The goal is to have the tick rate of a GPS satellite as observed from a receiver at sea level be equal to the tick rate of the best atomic clocks, adjusted to sea level. While this is the goal, it is not the reality. The smallish atomic clocks on the GPS satellites aren't as accurate as are the very large atomic clocks maintained by the US Naval Observatory. The GPS satellite atomic clocks are updated regularly, roughly once per day, to satisfy the promised accuracy of GPS receivers.
TAI is an ideal time scale as opposed to an operational time scale. To best represent this ideal, time as ticked by atomic clocks worldwide are averaged over the course of a year, weighted by the accuracies of those clocks. This addresses issues such as tidal effects and subtle changes in elevation. As the most accurate atomic clocks currently are those maintained by the US Naval Observatory, those clocks get more weight as to what constitutes TAI.
The end result is that GPS time and TAI are not separated by exactly 19 seconds. However, except for those who need milliarcsecond pointing accuracy, treating GPS time as being offset from TAI by exactly 19 seconds is more than good enough.