The distinctions between these two colloquial expressions are somewhat more complex.
The phrasal-prepositional verb "cut loose FROM" means "cut or sever ties with."
Although I can't find a source that says so, I'm going to go out on a limb and say that "cut loose WITH" is a phrasal-prepositional verb meaning "to emit or produce (sound or movement) without restraint." (I'm also going to say that there's a different verb, "cut loose," which will be described below.)
Here are a few examples from Google:
CUT LOOSE FROM
--Unless they can cut loose from their parents, these children may not be able to attach themselves to someone else in a marital relation?ship.
--This was a fun project to work on, as I got to cut loose from the standard comic strip format, and draw a story in glorious, full-sized comic book format!
CUT LOOSE WITH
--Then the congregation started to nervously laugh while watching us and when they gave up and cut loose with loud laughter, the minister started to snicker
--He laughed. He threw his head back and really cut loose with a mirthful, deep laugh.
(In American English--I don't know about British English--this same meaning is expressed with "LET loose with.")
--The youngsters let loose with a hearty round of applause as the approximately 10-minute-long contact [with the astronauts] ended)
I think that there's a phrasal verb, "cut loose" that has a different meaning, "behave without the ordinary restraints." It can occur alone, without anything following, as in
--St. Barts is the place where a woman can really cut loose and wear clothes that are too racy for home (at least for folks like me who live in the Bible Belt).
--Though he can cut loose when he wants to, Kevin's voice is soft and warm, and he's got a smooth delivery which the band deftly supports
When this verb is followed by the preposition WITH, it does not form a phrasal-prepositional verb; it is simply "cut loose" plus the ordinary preposition WITH. For example,
--With Mom safely snoozing on the other side of the island, the boys finally get a chance to cut loose with their friends and a big green hot rod!
--In 1831, Mme. Dudevant cut loose with her husband's consent and went to live in Paris.
--Every year, says the Royal Society for the Prevention of Accidents, 220,000 [do-it-yourself] dare-devils turn up in need of treatment for cuts and breaks, falls and concussions. ... So before you cut loose with a high-speed drill or start slashing about like a Samurai with a razor-sharp Stanley knife, take some time planning the job ....
So if my claims are valid, the definition of "cut loose WITH" needs to be emended.