as long as the thread you choose is strong (doesn't break when you give it a tug) fits your needle & is the right color you want to use - you can use what ever thread you want to piece with- some people like a thin/fine thread (like 50-60wt thread for piecing- because it is fine-does not distort the seams- some people like 30 or 40 wt- personally i piece with silk thread- but that is just my choice-
hand quilting thread is often a heavier thread- but does come in finer weights (the higher the wt number the thinner/finer the thread- so 30 wt thread is twice as thick/heavy as 60 wt)
hand quilting thread should never be used in a machine- it has a waxy coating on it to help it glide through the fabric by hand-which can gum up & cause damage to your machine.
if you visit the thread company sites (like Superior Threads) they have free charts that tell you about the different threads, what they are good for, the differences in needles, what needle to use with which thread- they are a very good resource- & free.