There are several main reasons for the rejection of the Apocrypha.

Non-acceptance by the Jewish canon.

The Jewish Canon does not include the Apocrypha. This is significant as it was to the Jews that the OT was entrusted (Rom 3:1,2) and they are the custodians of the limits of their own canon.  (Some of the Apocrypha books were written in Greek, not Hebrew).

The Jewish scholars of Jamnia (ca. A.D. 90) did not accept the Apocrypha as part of divinely inspired canon.

Philo, an Alexandrian Jewish teacher (20 B.C.- A.D. 40) quoted extensively from virtually every canonical book but never once quoted the Apocrypha as inspired.

Josephus (A.D. 30-100), a Jewish historian explicitly excluded the Apocrypha, speaking of the number of Jewish books which are divinely trustworthy, not leaving a place for the apocryphal books. . He numbered the OT books as 22 (the equivalent of the 39 books in the Protestant Old Testament). Josephus expressed the common Jewish perspective when he said that the prophets wrote from the time of Moses to that of Artaxerxes, and that no writing since that time had the same authority.

The Jewish Talmud teaches that the Holy Spirit departed from Israel after the time of Malachi, both of whom lived about four centuries before Christ, while the books of the Apocrypha were composed in the vicinity of two centuries before Christ.

There are several statements by Rabbis that prophecy ceased in the fourth century B.C. acknowledging that the Apocrypha was written in a period when God had ceased giving inspired writings.

Seeming Exclusion by Jesus Himself.

When Jesus or the apostles appealed simply to "the Scriptures" against their Jewish opponents, there is no suggestion whatsoever that the identity and limits of such writings were vague or in dispute.

Jesus seems to exclude the Apocrypha in his statement in Luke 11:51 - "from the blood of Abel to the blood of Zechariah who perished between the altar and the temple. Yes, I say to you, it shall be required of this generation" (NKJV).

Christ uses the expression "from the blood of Abel to the blood of Zechariah," The death of Abel is recorded in Genesis, the first book in the Hebrew canon. The death of Zechariah is included in 2Chronicles, which appears troublesome since Zechariah was not chronologically the last martyr mentioned in the Bible (cf. Jer. 26:20-23). However, Zechariah is the last martyr we read of in the Old Testament according to Jewish canonical order (cf. II Chron. 24:20-22), which was apparently recognized by Jesus and his hearers. The traditional Jewish canon was divided into three sections (Law, Prophets, Writings), and an unusual feature of the last section was the listing of Chronicles out of historical order, placing it after Ezra-Nehemiah and making it the last book of the canon. In light of this, the words of Jesus in Luke 11:50-51 reflect the settled character of the Jewish canon (with its peculiar order) already established in his day.

The order of books as they appear today is taken from the Septuagint (second century BC Greek translation of the OT).

Lack of reference to the Apocrypha in the NT.

While the NT quotes mainly from the Greek Old Testament (LXX) it is uncertain as to whether the Septuagint contained the Apocrypha. No direct quotations from any Apocryphal books appear in the NT although they were aware of these books and alluded to them at times. However Hebrews 11:37 may very well refer to 1 Kings 17:22 and not 1 Maccabees, as is often claimed. But none of these allusions rise to the apostles using the Apocrypha as an authoritative source. On the other hand there are literally hundreds of quotations in the NT from the ‘Law and Prophets’ which Jesus called “all the Scripture”. Luke 24:27).

So with the lack of authoritative quotes from the Apocrypha in the NT, it appears the NT writers, and Jesus Himself, did not accept the Apocrypha as Scripture.

Not every book of the Hebrew canon is quoted in the NT (such as the Song of Solomon, or Canticles as it is named in Catholic Bibles). But every section of the OT as Jews divided it is quoted from (i.e. the Torah, the Prophets, and the Writings).

The absent of any quotes from any of the Apocryphal books is rather striking. This is especially so given that many of the OT quotes in the NT were actually taken from the Septuagint and not from the Hebrew text (It is for this reason that if you compare OT quotes in the NT with their OT counterparts they don't match up exactly). In any case, what this means is, the Bible text the NT authors had before them had the Apocrypha in it; but they seem to have completely ignored the Apocrypha when they were looking to support a statement with an authoritative source.

So the evidence of the NT strongly suggest the writers of the NT did not accept the Apocryphal books as canon.

Rejection by many early church fathers.

Early church fathers like Origen, Cyril of Jerusalem, Athanasius, and The great Roman Catholic translator Jerome spoke out against the Apocrypha.

In 382 Bishop Damascus had Jerome (the greatest Bible scholar of the early Medieval period) work on a Latin text to standardize the Scripture. The resulting Vulgate was used throughout the Christian world though Jerome himself separated the Apocrypha from the rest. He stated the church reads them “for example and instruction of manners”, but does not “apply them to establish any doctrine”. More damning was his statement that “they exhibit no authority as Holy Scripture” (Preface to Vulgate Book of Solomon,) He initially refused to translate the apocrypha into Latin but later made a hasty translation of a few books.

But Augustine (c. 400 AD) did recognize the Apocrypha. So it was mainly from his influence that the Apocrypha eventually became accepted. However, the Catholic Church itself did not officially canonize the Apocrypha until the council of Trent in the 1500's. (Called as a response to the Protestant heresy by the Catholic Church) Before that, there was always debate as to their inspiration. Some of the Church Fathers, for instance, quoted from the Apocrypha as Scripture but others didn't. But however it is looked at, the official acceptance of the Apocrypha occurred well after the NT was written and the final canonization of the Hebrew Scriptures by the Jews.

The reformers were also forced to face the Canon issue. After the Reformation the books of the canon were widely agreed on. Instead of the authority of the Church, Luther and the reformers focused on the internal witness of the Holy Spirit.

Test of Propheticity.

One of the tests of whether writing was seen as inspired or not was propheticity. God determined which books would be in the Bible by giving their message to a prophet. Only books written by a prophet or accredited spokesperson for God belong in the canon. The people to whom the prophet wrote knew which of the prophets fulfilled the requirements for god’s representatives and authenticated the writings by accepting them.  Moses’ books were accepted and stored in a holy place (Deut 31:26) as was Joshua’s writings (Joshua 24:26). Daniel already had a copy of Jeremiah and Peter had a collection of Paul’s writings, equating them with the OT as Scripture. False prophets were weeded out if their prophecy did not come true (Deut 18: 20-22) and alleged revelations that contradicted previously revealed truths were rejected as well. (Deut 13:1-3.).

Additionally Moses’ writings are cited through the OT beginning with Joshua all the way to Malachi. (E.g. Josh 1:7, 1 Kings 2:3, Ezra 6:18, Jeremiah 8:8, Malachi 4:4.

Later prophets cite earlier ones (E.g. Jer 26:18, Ezek 14:14, Dan 9:2, Jonah 2: 2-9).

On the NT Paul cites Luke (1Tim 5:18), Peter recognizes Paul’s Epistles (2 Peter 15-16)

Revelation Is filled with imagery largely from Daniel.

On the other hand no Apocryphal book claims to be written by a prophet and there is no predictive prophecy in the Apocrypha. Not once is an Apocryphal book cited authoritatively by a prophetic book written after it, nor is there any supernatural confirmation of the writers of the Apocrypha as there is for prophets who wrote the canon.

Errors in the Apocrypha

The books of the Apocrypha abound in doctrinal, ethical, and historical errors. For instance, Tobit claims to have been alive when Jeroboam revolted (931 B.C.) and when Assyria conquered Israel (722 B.C.), despite the fact that his lifespan was only a total of 158 years (Tobit 1:3-5; 14:11)! Judith mistakenly identifies Nebuchadnezzar as king of the Assyrians (1:1, 7). Tobit endorses the superstitious use of fish liver to ward off demons (6: 6,7)!

The theological errors are equally significant. Wisdom of Solomon teaches the creation of the world from pre-existent matter (7:17). II Maccabees teaches prayers for the dead (12:45-46), and Tobit teaches salvation by the good work of almsgiving (12:9) -- quite contrary to inspired Scripture (such as John 1:3; II Samuel 12:19; Hebrews 9:27; Romans 4:5; Galatians 3:11).

In Summary:

 They are not, and have never been, in the Jewish canon.

 Josephus explicitly excluded them from his list.

 Philo (20 B.C.-50 A.D.) neither mentions or quotes them.

 They were never quoted or alluded to by Jesus Christ or any of the apostles. The sermons in the Book of Acts, which outline Jewish history, do not included apocryphal events.

 Jewish scholars meeting at the Council of Jabneh did not recognize them.

 Most Church Fathers in fact rejected them.

 None of the Apocrypha claim inspiration or divine authority.

 Many of the Apocryphal books contain historical, geographical, and chronological errors.

 Many of the Apocryphal books teach heresy, contrary to the Word of God.

 Their literary style is legendary and fantasy. Some stories are grotesque and demonic.

 They lack the power and distinctive elements of the Word of God.