Source: M. Gilbert, "Wisdom Literature," Jewish Writings of the Second Temple Period, ed. Michael E. Stone (Philadelphia: Fortress, 1984) 300.
On Ben Sira:
"In the first century C.E. the book continued to circulate, as at Masada and Qumran. But nothing is known about the use made of it or the interpretation it was given in Judaism. There is no trace of its canonicity being debated at the synod of Jamnia (Yavneh) at the end of the century. One even has the impression that no objections were raised to its not being part of the Jewish canon of scripture. The opinion of Rabbi Akiba (d. after 135 C.E.) is of great importance. According to P.T. Sanhedrin 10, 28a, Akiba held that the book of Ben Sira was one of the 'exterior books, the reading of which excludes from the world to come.' According to Moore, the 'exterior books' would be the equivalent of 'the books of the minim', meaning those of heretics, as can be seen from B.T. Sanhedrin 100b, and probably according to T. Yadaim 2:13, Christian books. But Ginsberg has shown that 'exterior books' are simply 'books which are not part of the canon of scripture'. According to Rabbi Akiba, they could not be used for public reading or for school instruction. But Ben Sira could be read in private. And the many quotations from it given by the Tannaim and the Amoraim show that it was highly esteemed."
Ben Sira may have considered himself divinely-inspired. In Sirach 24:33, he says that he will, "pour out teaching like prophecy, and leave it to all future generations" (NRSV).
I'm not sure why some believed he belonged in the biblical canon, or why others did not. I've learned at least one thing from my study of the deutero-canonical writings: the post-apostolic fathers disagreed about the status of what Protestants call the "apocrypha." And scholars usually seek ideological reasons that Jews rejected the deutero-canonical writings: Ben Sira, for example, doesn't believe in an afterlife.
But there are books that made it into the Jewish canon, and they don't really believe in an afterlife either. Job is one example. Ecclesiastes is another. And, as we saw in Uninspired Canonical Books?, rabbis debated about Ecclesiastes' inspiration.
Some think that the deutero-canonical writings were excluded because the Christians used them. That's what a Catholic apologist said in a debate with James White. Maybe, but I don't see how the "apocrypha" supports Christian doctrine more than other books of the Hebrew Bible.
One professor told me years ago that the problem was their language: they weren't in Hebrew or Aramaic. But some of them had Hebrew or Aramaic originals (i.e., Ben Sira, possibly Tobit), so language can't be the only factor behind their rejection.
I've read somewhere that the rabbis closed the canon with Ezra, implying that they viewed Ezra as the last prophet. And that fits the Jewish canon quite well. The Hebrew Bible does not contain a prophet after the time of Ezra. While liberal scholars date Daniel to the second century B.C.E., the book presents itself as exilic, and that's probably what the rabbis went with when they included it in the canon. And Jewish tradition affirms that Ezra wrote Chronicles, the last book of the Hebrew Bible.
Why did Christians accept the deutero-canonical writings? I'm not sure. Maybe it was because Jews in the diaspora embraced them (for whatever reason). Or perhaps they weren't eager to close the canon with Ezra, since they believed that prophecy still continued. After all, they had their own set of new biblical books, which post-dated Ezra by over five centuries. We call it the New Testament. So why wouldn't they believe that books after Ezra could be inspired?
The Palestinian Jews, however, may have desired a clear cut-off point because of the instability of prophecy. As a professor at Hebrew Union College once said, prophets create instability, since they can claim a divine revelation that undermines the social structure. And, in the time of the Romans, the Jews felt they didn't need prophets who continually instigated political revolution. That may be why they made Ezra the cut-off man.