The title is the question. Supposedly, Andy Gibb never performed with his brothers, the Bee Gees. But why not? 
Family dispute? Amicable differences in musical style? What gives? - Jinx
Andy Gibb was significantly younger than his brothers. If I recall correctly, the Bee Gees had their first American hit when Andy was 10 or so.
Andy was born in 1958, Barry in 1946, and Robin and Maurice in 1949. According to this discography, Andy turned seven years old the year Barry Gibb and the Bee Gees Sing and Play was released (1965).
I, too, understood Andy to be too young at the time. But I always wondered, with the falsetto songs recorded, why they didn’t bring him in anyway, ala Michael in the Jackson 5
Andy didn’t simply want to be known as “The Bee Gees’ little brother,” and they were already quite famous by the time he was in his late teens, so he embarked on his own career. He apparently spent much of his adult life torn between the desire to be independent and to have the support and approval of his brothers. It’s been suggested that this internal conflict is what drove him to drink and use coke to excess until he died of drug-related heart ailment at the age of 30. IIRC he actually always had a good relationship with his family otherwise, and he and Barry were quite close.
The Bee Gees themselves started performing as The Brothers Gibb when Barry was just 12 and the twins were nine. Throughout their early career they did NOT sing with falsettos, but rather did mostly simple songs with a fair amount of harmonizing. The falsettos only really emerged with the release of "You Should Be Dancin’ " from Children of the World in 1976, and took off with the Saturday Night Fever soundtrack in 1977. (Just for fun, compare the sound clips from Main Course, released in 1975, with clips from Children of the World.
Andy hit the pop scene as a solo artist shortly thereafter. Maybe it would have been possible for him to join The Bee Gees before SNF, but I don’t think there is really any way he could have joined after without being accused of shoehorning in on his brothers’ success. (Think of that poster with The Bee Gees dressed in white spandex - it was a striking pose of THREE people. How would you later fit in another such that he looked like he belonged?) Besides, although Andy was capable of singing falsetto, IMHO he never did it as well as Barry could; I don’t remember him ever singing an entire song that way.
As far as Andy performing with his brothers - I have some vague recollection that they sang a song or two together on stage once, but that was about it. Barry sang back-up for Andy on his albums, but then, in those days, Barry sang back-up for a lot of people (including Kenny Roger and Dolly Parton on Islands in the Stream!).
Andy could very well continued to have a successful solo career. It’s a pity his inner demons won control and he lost his life.
[sub]I suppose this is as good a place as any to admit that my first-ever concert was Andy’s show at the Nassau Coliseum ca. 1978. 
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I just want to mention that Andy Gibb had a fantastic run in “Joseph and the Amazing Technicolor Dreamcoat,” and could have had a good career in musicals.
When I was twelve years old I sat in the 10th row of the Pontiac Silverdome and saw the Bee Gees LIVE. I had the time of my life. When Andy came out and sang “You Should Be Dancin’” with them for a final encore I thought I’d died and gone to teenybopper heaven.
He performed with them. Several times.
In an interview, Barry said that when Andy started getting under the influence of the wrong crowd in Hollywood, he wanted to bring Andy back into the family fold and make him an offer to join the group. Said he was voted down 2 to 1, which means that that Maurice and Robin voted no. He didn’t give a reason, but there were several valid reasons, one being that they had enjoyed a great deal of success without Andy, another being that they had paid their dues together while Andy hadn’t. Then there’s the teeny bopper factor, which was probably the single most harmful thing that happened to the group. Why add another “heart throb” to the band and risk even more backlash?
Robin later said that he didn’t think Andy was interested in continuing to pursue a career in music, but that Andy never did have a game plan of what he did want to do. IMO, once he became a “teeny bopper” it limited his career path enormously. Not only did he suffer the BeeGees backlash which limited his music career, but had he decided to become a plumber, who wants to be known as a former teeny bopper turned plumber? Either way, he was in for ridicule.