“wear off” is used for things that are applied topically:
"The paint was worn off the top of the hand rail."
"Cheap nail polish wears off quickly.”
“This suncreen will wear off after only 1 hour in the water, so re-apply frequently.”
“The letters on my kyboard are worn off, but the keys themselves aren’t worn out yet."
Or for the effect of something that someone has been exposed to (a drug or chemical):
"The effects of mustard gas don’t wear off. If the victims survive the attack, the effects of the gas will last for the rest of their lives.”
“The drug wears off after 3 hours, so you will be able to drive home yourself.”
“wear out” is not acceptable in those circumstances.
“wear out “ is used when something is not usable anymore: clothes, cars computers, books, stoves, etc:
"This book is worn out. The print is so worn off I can’t read it anymore.”
“My hip is worn out, I can barely walk."
Or when people are tired or exhausted physically or emotionally:
“I’m worn out from scrubbing the floor all day.”
“Going through that divorce really wore him out."