Why did Albus Dumbledore hide the identity of Aberforth Dumbledore?

Question

What happened between Aberforth and Albus? And why did Albus hide both his brother's identity and that of his sister Ariana?

Answer 1

(Harry and Doge discussing Dumbledore during Bill and Fleur's wedding):

'I saw the obituary you wrote for the Daily Prophet,' said Harry. 'I didn't realise you knew Professor Dumbledore so well.'

'As well as anyone,' said Doge, dabbing his eyes with a napkin. 'Certainly I knew him longest, if you don't count Aberforth - and somehow, people never do seem to count Aberforth.'

Harry Potter and the Deathly Hallows - p.127 - Bloomsbury - Chapter 8, The Wedding

(The Life and Lies of Albus Dumledore):

'He were a headcase, that Aberforth,' says Enid Smeek, whose family lived on the outskirts of Godric's Hollow at that time. 'Ran wild. 'Course, with his mum and dad gone, you'd have felt sorry for him, only he kept chucking goat dung at my head. I don't think Albus was fussed about him, I never saw them together anyway.'

...

'Aberforth blamed Albus, you know, as people will under these dreadful circumstances. But Aberforth always talked a little madly, poor boy.'

Harry Potter and the Deathly Hallows - pp.289 and 292 - Bloomsbury - Chapter 18, The Life and Lies of Albus Dumledore

(Doge on Aberforth in his obituary):

Three years after we had started at Hogwarts Albus's brother, Aberforth, arrived at school. They were not alike; Aberforth was never bookish and, unlike Albus, preferred to settle arguments by duelling rather than through reasoned discussion.

Harry Potter and the Deathly Hallows - p.22 - Bloomsbury - Chapter 2, In Memoriam

(Dumbledore to Harry at King's Cross):

'And then ... you know what happened. Reality returned, in the form of my rough, unlettered, and infinitely more admirable brother.'

Harry Potter and the Deathly Hallows - p.574 - Bloomsbury - Chapter 35, 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.

Answer 2

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.

The little people in the photograph jostled among themselves and those hidden right at the back appeared at the forefront of the picture. ‘That’s Dumbledore’s brother Aberforth, only time I ever met him, strange bloke … that’s Dorcas Meadowes, Voldemort killed her personally … Sirius, when he still had short hair … and … there you go, thought that would interest you!’

^{Harry Potter and The Order of the Phoenix | Chapter 9: The Woes of Mrs Weasley}

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.