What happened between Aberforth and Albus? And why did Albus hide both his brother's identity and that of his sister Ariana?
Albus didn't divulge that the barman of the Hog's Head was Aberforth, but he didn't deny it either.
We do know Aberforth was in the Order of the Phoenix so they must have had a later cordial relationship, and Moody at least knew he was Dumbledore's brother.
He also never hid the fact he had a sister, he just wasn't asked.
Harry, in Harry Potter and the Deathly Hallows, comes to the realisation that in all his meetings with Dumbledore he never once asked him about himself and his family, and as we only see Dumbledore through the eyes of the Harry we don't get to learn his past.
Aberforth blamed Albus for the death of their sister. That's why they fell out.
(Harry and Doge discussing Dumbledore during Bill and Fleur's wedding):
(The Life and Lies of Albus Dumledore):
(Doge on Aberforth in his obituary):
(Dumbledore to Harry at King's Cross):
I think the answer is simple. No one cared about Aberforth, no one was interested, no one asked him about the uninteresting, unsuccessful, uncouth, unintelligent goat botherer. Who's gonna ask Albus Dumbledore, in interview, about his unglamorous brother, when they could be talking about the twelve uses of dragon's blood, or the hot political topic of the day, or his latest decision as Chief Warlock of the Wizengamot?
Obviously this is entirely unfair on Aberforth, but I think that was one of JKR's points. The answer is simple: nobody cared. Which turned out to be quite helpful with the Order and then, later, after Dumbledore's death. Aberforth, unnoticed, could have the mirror and look after Harry in Hogsmeade and all of that. And I think, partly, Dumbledore might have been quite happy about that and that might have contributed to his not bringing Aberforth up - especially to Harry. And, obviously, after Ariana, I think it's pretty obvious why Aberforth wouldn't have wanted to define himself as Dumbledore's brother and wouldn't have paraded the fact, and so he went pretty well unnoticed.