Why did Gregory and Maurice Hines break up their two-man act?

They didn't break it up, their father joined them and the act became Hines, Hines and Dad. Gregory left the group in 1973 to form a jazzrock group called "Severance".

Hines, Hines, and Dad was a great trio but after the dad left, the two brothers had a falling out. What was that about?

I'm only finding this:

Hines’s relationship with his brother was also strained, and they often did not speak to each other. Hines knew the partnership was over when he and his brother Maurice had an argument that nearly ended in blows.

http://encyclopedia.jrank.org/articles/pages/4295/Hines-Gregory-1946-2003.html

With older brother Maurice, he'd been tap dancing professionally since he was five. The Hines Kids became the Hines Brothers and, when Maurice Sr. joined the act in 1964, became Hines, Hines & Dad. By this time--green, unseasoned and barely old enough to vote--he'd fallen in love and married. And now he felt smothered, trapped by what his life had become. But freeing himself meant hurting--leaving--all the people he cared about most.

Painful as it was, Hines soon after broke up his marriage and the family act and moved to Venice Beach, Calif. He had no job and no plan--just a serious taste for sex, drugs and rock `n' roll--in that order. With $5,000 he'd borrowed from Bill Cosby, he started a jazz-rock band and scraped out a living "playing guitar and singing in bars around L.A. for five bucks a night and all the beer we could drink."

http://tinyurl.com/j2ajgub

I don't believe there was a "falling out" per se. From Gregory Hines' online biography, the following can be found:

"Gregory and Maurice then grew into the Hines Brothers. When Gregory was eighteen, he and Maurice were joined by their father, Maurice Sr., on drums, becoming Hines, Hines and Dad. They toured internationally and appeared frequently on The Tonight Show, but the younger Hines was restless to get away from the non-stop years on the road, so he left the group in his early twenties and "retired" (so he said) to Venice, California. For a time he left dancing behind, exploring alternatives that included his forming a jazz-rock band called Severence. He released an album of original songs in 1973."

https://www.nypl.org/node/70548

This seems to indicate that their professional paths just took different courses. The brothers appear to have reunited several times in later years (before Gregory's untimely death at 57), as shown by the following quote from the same online biography:

"Vincent Canby in The New York Times wrote about [Gregory] Hines' rare screen presence in the film: "He doesn't sneak up on you. He's so laid back, so self assured and so graceful, whether acting as an ambitious hoofer or tap dancing, alone or in tandem with his brother, Maurice, that he forces YOU to sneak up on HIM."

Whatever the reason for their professional divergence, Maurice Hines now remembers his brother fondly and has a one-man show detailing the brothers' early career together. It is called "Tappin' Thru Life", and it's a fantastic show and I highly recommend it.

http://www.tappinthrulife.com/

In Maurice's own words, this is how he felt about his work with his brother:

"Maurice Hines is Tappin' Thru Life started with my wanting to celebrate my brother Gregory Hines. I read an article on tap a few years ago and his name wasn't mentioned once - SO WRONG! Now I get the chance to honor him properly on how fabulous a tap dancer he was. I also wanted to thank the wonderful performers ELLA FITZGERALD, FRANK SINATRA, LENA HORNE, NAT 'KING' COLE and JUDY GARLAND who were all an inspiration to a young performer on how wonderful being on stage could be. My main focus for anytime I'm on any stage is to follow what my wonderful mother told Greg and I... that no matter what you do on stage always do it with CLASS and make sure the audiences leave saying WOW! I HAD A GREAT TIME, BABY!"
-- Maurice Hines

http://mauricehines.com/tappin.html