This is an attempt to answer the Frequently Asked Question: why
do Protestants reject the Apocrypha?

I think it has to do with the different status of the Bible for
Protestants.  For Catholics and Orthodox, the Bible is a repository of
Tradition, but is far from the only source of authority: the Church is
able to make authoritative interpretations of Scripture.  However in
the Protestant tradition, Scripture is the preeminent authority.  This
means that Protestants have to be a bit more careful about what they
allow to be Scripture.  Consider for example Augustine.  He is quoted
by Calvin as saying "The Jews do not consider the writing of the
Maccabees as the Law, Prophets, and Psalms, to which the Lord attests
as to his witnesses, saying: 'Everything written about me in the Law
...  and the Prophets and the Psalms must be fufilled.' [Luke 24:44].
But it is not unprofitably received by the church if it be soberly
read or hearkened to."  This is not a problem for Catholics, as all of
Scripture is interpreted by the Church.  But for Protestants, there
can't be things in Scripture which are questionable.

Anyway, here are some of the standard arguments:

- Christ refers to the Law and Prophets, and often quotes from the OT.
The Apocryphal books do not fall into these categories.  While people
speak of allusions to them in the NT, there are no actual quotes from
any of the Apocryphal books in the NT.

- even in the early church, there was sometimes a distinction made
between them and the Hebrew OT.  This seems to have started in the 4th
Cent.  Several Greek fathers (Eusebius, Athanasius, Cyril of
Jerusalme, Epiphanius, Gregory of Nazianzus, according to the Oxford
Dictionary of the Christian Church), while still quoting them a
Scripture, regarded them as of a lower class.  Jerome is the main
Western authority who followed this approach.  While generally the
West continued to read them as Scripture, there were a few that
continued to question it, apparently based primarily on Jerome's
assessment.  Calvin cites Cyprian [actually, a work attributed to
Cyprian by Erasmus but about whose authorship Calvin has doubts -- the
editor of the Institutes cites it as by Rufinus, a 4th Cent writer] as
evidence that the early church didn't regard the Apocrypha as
canonical.  [His statement seems to go beyond the evidence: while some
distinctions were made, they were not as universal as he implies.
However libraries in the 16th Cent weren't as extensive as now, and
Calvin may have honestly believed that the 4th Cent. doubts were
representative of the early church.]

- the Reformation was strongly influenced by the humanist credo "ad
fontes" [back to the original sources].  Thus scholarship was done in
the original languages rather than the Latin Vulgate.  The Apocryphal
books are not present in the Hebrew, and were not recognized by Jews.

- proof texts for several doctrines that the Reformers considered
dubious was found in the Apocrypha (e.g. praying to saints and
Purgatory).

- Calvin is bothered by the fact that the author of Maccabees does not
seem to regard his work as authoritative.  He cites II Mac 15:39.

Note however that the original Reformers were rather more moderate in
their views on the Apocrypha than is often realized.  Luther
translated them as part of his Bible, and said that they are "good and
useful to read" (words rather similar to Augustine's).  The Geneva
translators also included them, with wording similar to Luther's.  The
Westminster Confession took a somewhat harder line, saying that they
had no more status than other human writings.  Carlstadt called about
half of them "holy books", but with a lower status than canonical.  At
least in English, it appears that omitting them from printed editions
of the Bible was due to a demand for smaller Bibles (though obviously
it was influenced by the understanding that these books had less
authority than the others).