"Worry" is the common, everyday word. "Fret" is rare.
To me, "worry" suggests anxiety that is focussed on one single, fairly important thing. "Fret" suggests one of many small, unimportant worries.
"I am worried about the surgery I need to have."
"I'm fretting about all the things I need to do tomorrow. I have to mail a letter, buy groceries, do my laundry, get my hair cut, and deposit a check at the bank."