What is the difference between Hobbs Time and Tach Time?
Hobbs Time

In most planes, the Hobbs clock is started and stopped based on an oil pressure switch, so it starts when the engine starts, and stops when the engine is shut-down. While it's running, it just ticks off a tenth of an hour every 6 minutes, based on "regular wall clock time". So a tenth of idling on the ramp is the same as a tenth at cruise.

Tach Time

The tach clock isn't really a clock at all, it doesn't actually measure time, it really measures engine revolutions. But it's calibrated such that a tenth of an hour of tach time is clicked off when the engine is at cruise RPM for 6 minutes. In other words, if the plane is at cruise RPM, the tach clock will be clicking off tenths of an hour at the same rate as the Hobbs clock, and the same as the watch on your wrist. But if the engine is idling at an RPM speed that's half of what cruise RPM is, then the tach clock will be running at half the speed of the Hobbs clock.

Relation

So, for the tach clock, less "time" is clocked when the plane is idling on the ramp, or flying at low RPM. For short flights (where ramp idling time is a significant percentage of total time), and flights where you're doing a lot of pattern work (and thus operating at low RPM), tach time will be significantly less than Hobbs time.

Maintenance

When an engine is replaced or overhauled, that engine is considered to be "zero time". Whether they actually reset the tach gauge to 0 when the engine is overhauled, or if there's a way to put a time on a brand new tach or Hobbs meter when the old one dies and is replaced, I dunno, but in general, I'd assume that the aircraft's engine and airframe log books are really the official recording of the progression of airframe/engine times, so I don't know that it's particularly important whether the tach is set to the engine's time, or if it's really just used as an incremental meter.



AOPA has a nice (and long) article about this.