Putz can either be a noun or a verb. As a verb, it is the act of wasting time or being aimless.
Futz is only a verb. It also means to wander without direction or spend hours on pointless endeavors. It is most often seen paired with the preposition around.
So, futz and putz are synonyms as verbs. However, they do not have the same origin.
Putz comes from the Yiddish word putz meaning penis.
Futz does not have a clear origin, but the educated guess made by most dictionaries is a variation on the Yiddish words arumfartsn zikh meaning to fart around.
Fart around is listed in most dictionaries as another synonym of putz and futz. Outside the United States it is more commonly found as fart about.
When he is home, he spends his time “putzing around in my studio,” he said. [Chicago Tribune]
Instead of having to futz with SD cards or USB thumb drives, you can send photos to the frame wirelessly from your computer or smartphone. [The Wall Street Journal]
That’s how long it’s been since Aussie duo the Veronicas last released an album, an unplanned furlough caused by record company futzing over their musical direction. [The Guardian]
“I tell you, we are here on Earth to fart around, and don’t let anybody tell you different.” [The Telegraph]
“Miller and I then farted about improving our golf handicap for a couple of years,” says Roper. [The Independent]