The offset is changed as needed to keep UTC within about 0.7 seconds of earth rotation time, UT1 . Terrestrial Dynamic Time, which is tied to atomic time by a constant offset of 32.184 seconds, replaced ET at the beginning of 1984. The purpose of the offset is to maintain continuity between ET and TDT at the transition. There is a subtle relativistic distinction between coordinate time and dynamic time, which is not significant for most practical purposes. The rate difference from TT can be important to long term measurements, so make sure you know which time is being used when comparing observations.
Some physical constants are different in coordinate time.
You are not likely to encounter TCG in the literature. These corrections amount to as much as about 1.6 millisends and are periodic with an average of zero. The dominant terms in this correction are have annual and semi-annual periods. There is a subtle relativistic distinction between coordinate time and dynamic time, which is not significant for most practical purposes. The rate difference from TDB can be important to long term measurements, so make sure you know which time is being used when comparing observations.