Cauterizer give you instant pressure with its attack but usually cannot be clicked after a few turns, because the Engineers it requires will have died in defense. At this point it can still be used to block. As such Cauterizer should most of the time be thought of as a defensive unit rather than a priority buy, and should not be rushed.
Cauterizer is most useful as a unit which counters mass chill strategies such as Frost Brooder and Endotherm Kit because of the Cauterizer's ability to have four Engineers on blocking while it attacks and is a good blocker even after you run out of Engineers to power it.
Cauterizer has good synergy with units that produce Engineers such as Hellhounds (which are also blue-red units) and Sentinels as they can replace Engineers lost on defense and keep Cauterizer firing. It also combines well with Antima Comet as you will be producing/buying Engineers for Antima Comet and have a large supply of energy to power the Cauterizer. Tesla Coil generally works against Cauterizer as it sacrifices Engineers to fire.