Handy is a very general word… So a man can be handy - he knows how to fix various little things around the house, no specification. Hence the compound noun handy+man= handyman. Someone is adept AT something …..it is very specific. Adept can be adjective and noun too. Meaning expert like, excellent He is an adept at tiling. He is adept at tiling. Deft could be a synonym for adept. It is the same meaning = expert like but also fast.

I can fix it myself. I’m pretty handy. (Not deft, adept) I’m adept at getting old cars running again. It’s how I make my living. (Not deft, handy) ‘Deft’ is rarer and usually used as an adverb. She deftly assembled the parts of the machine, and it worked the first time she turned it on.