What is the difference between "umgezogen" and "verzogen"? Or is it just the same? Especially "umgezogen nach" and "verzogen nach" seem to be exactly the same. Just the combination with "unbekannt" seems to be exclusively used with "verzogen".
With regard to moving, there's no difference at all. Both verbs convey the idea of changing your address.
Verziehen, however, is rarely used; only occurs in the form verzogen sein or unbekannt verzogen.
You're always on the safe side using umziehen (or wegziehen, fortziehen, meaning to move away).
Both verbs have more meanings, but there's no further overlap.
1 Wir sind nach München umgezogen - meaning we have moved house and live now in Munich.
2 If a letter is turned back to the post office with the remark "verzogen" this means the addessee has moved house, but his new address is unknown.
3 ein verzogenes Kind has no manners because the parents don't know anything about child-raising.
In German ziehen (to pull) is used as a stem for a variety of verbs relatied to moving house. Of course the meaning ziehen is not restricted to moving alone, sometimes the same prefix is used with an entirely different meaning:
umziehen: to move (house) also to change clothes
wegziehen: to move away also to draw away
ausziehen: to move out also to take of clothes
einziehen: to move in also to feed in, to infiltrate, to contract
zuziehen: to move to also to pull tight
verziehen: only used in the past with participle ist/war verzogen: moved away also spoiled, spoilt, skewed
umherziehen: to tramp, to rove around
nachziehen: to partly move later (e.g. family) also to (re)tighten, to limp ("ein Bein nachziehen")
beziehen: to move into also to obtain, to draw, to drap ...
herziehen, fortziehen, zusammenziehen, ... and likely many more - also see other answers
I would use "umgezogen" if i knew where exactly they have have moved. otherwise, i would use "verzogen". If i know the city or region, i might say "verzogen nach [region/city]".
I think "verzogen" therefore can also express that you are not willing to tell (the person who asked) their new adress.
On the other hand, when my answer is simply "Sie sind umgezogen", you can suggest that i also know where they moved, and ask me for the adress in your next question.