Why was Tosefta ("Supplement") not incorporated into the Mishnah? Is it any less authoritative? Many teachings therein are also in the Mishnah, but not all. Are those that are not included specifically rejected?
The Talmud in Sanhedrin 33a states:
רב ששת אמר כאן שטעה בדבר משנה כאן שטעה בשיקול הדעת דאמר רב ששת אמר רב אסי טעה בדבר משנה חוזר טעה בשיקול הדעת אינו חוזר א"ל רבינא לרב אשי אפילו טעה בר' חייא ורבי אושעיא א"ל אין אפילו בדרב ושמואל אמר ליה אין אפילו בדידי ודידך א"ל אטו אנן קטלי קני באגמא אנן
R. Shesheth said: Here it treats of a case where he [the judge] erred regarding a law cited in a Mishnah; there, of a case where he erred in the weighing of [conflicting] opinions. For R. Shesheth said in R. Assi's name: If he erred in a law cited in the Mishnah, the decision is reversed; if he erred in the weighing of [conflicting] opinions, the decision may not be reversed.
Rabina asked R Ashi: Is this also the case if he erred regarding a teaching of R. Hiyya or R. Oshaia? — Yes, said he, And even in a dictum of Rab and Samuel? Yes, he answered. Even in a law stated by you and me? Are we then reed cutters in the bog? he retorted.
Rashi there explains:
האי דבר משנה דקאמר מי אמרי' דאפי' טעה בדבר המשנה בתוספתא דבי רבי חייא ורבי אושעיא שהם סידרו התוספתא וסתמו בה מה שסתמו ולא הוי עיקר כל כך כמשנתנו שסתמה רבי שהיה גדול מהם וכל הישיבה מצויה אצלו על ידי שהוא נשיא ודקדקו עמו בדברים והעמידום על בוריים
This "matter of a Mishnah" that was said, do we say that it [applies] even [to] one who erred in a matter of a Mishnah in the Tosefta of R. Chiyya and R. Oshaya? [They] organized the Tosefta and finalized in it that which they finalized, and it was not as important as our Mishnah which Rebbe finalized. For [Rebbe] was greater than them and the entire Yeshiva was around him since he was the president, and they were meticulous with him about things and they established them [resulting] in their clarity.
This sounds like it's authoritative but not quite like a Mishnah.
In Chullin 141a we find that Beraitot that didn't make the R. Chiyya and R. Oshaya's cut are not as authoritative as those that did:
אמר ליה ר' זירא לאו אמינא לכו כל מתניתא דלא תניא בי רבי חייא ובי רבי אושעיא משבשתא היא ולא תותבו מינה בי מדרשא
Thereupon R. Zera said to them, Have I not told you that every Baraitha that was not taught in the school of R. Hiyya and R. Oshaia is not authentic, and that you should not put it forward as a refutation in the Beth Hamidrash?
First, I want to stress that I found all of this information in the Artscroll "Introduction to the Talmud" (at this point in time, a somewhat newly published book) which has lots of really great historical information on how the Talmud and it's sources were mesudar. I'd highly recommend it to answer these types of questions.
In the back of the book, they have beautifully translated the Iggeres of Rav Shreira Gaon (one of the prominent Gaonim ~900CE), where he answers in great detail many questions posed from Karaites about the veracity of the Oral Torah and its origins. There is a section where he discuss "The Mishnah Over the Tosefta":
[Now, it should be pointed out] that] if R' Chiya disagreed in a Beraisa with Rebbe, we do not listen to him. For example, Rebbi decided a law and registered it in a Mishnah without attributing it to any Tanna, although it had been a subject of dispute between Tannaim. If R' Chiya subsequently shows [in the Tosefta] that a disagreement had indeed taken place, even though the Mishnah does not show the entire picture, the final halachah is nevertheless decided in accordance with the Mishnah. We will not follow [in this case] the usual rules of deciding between different Tannaim, such as: In a dispute between R' Meir and R' Yose the halacha follows R' Yose (Eruvin 46b). Rather, the halachah is decided according to R' Meir, if his is the sole opinion mentioned by Rebbi in the Mishnah. However, we do follow the aforementioned rules when the Mishnah itself mentions the different opinions. Thus, where there is a dispute between R' Meir and R' Yose [explicitly mentioned in the Mishnah], the halachah would be decided according to R' Yose.
Additionally, if R' Chiya held that the opinion of R' Meir, which is recorded in the Mishnah as having been opposed to that of R' Yose [and therefore not accepted as the final halachah], was the correct one, and he thereby presented it in the Tosefta as the sole opinion, we do not accept R' Chiya's position.
Since R' Chiya was the disciple of Rebbi, whatever Rebbi taught him is what [R' Chiya] would say, and R' Chiya had only that which Rebbi had.
In fact, strictly, more weight is given to an inference from a Mishnah than to another opinion explicitly presented in a Baraisa [i.e. Tosefta].
He then points out that there is a preference given to R' Chiya and R' Oshaya's Beraisos over other collections of Baraisos, since there were errors and inaccuracies in the others.
Just some background info from some other sources:
Note that there are known to be two Tosefta's: One from the Tanna R' Nechemyah (learned straight from R' Akiva), which is not the same as the Tosefta we have in the back of our Gemara. The one in the back of our Gemara is the Tosefta of R' Chiya (the first of a number of R' Chiya's).
Regarding R' Nechamya's Tosefta, the Gemara in Sanhedrin 86a says:
סתם תוספתא ר' נחמיה
An unattributed baraita in the Tosefta [is in accordance with the opinion of] Rabbi Neḥemya
Regarding R' Chiya, the composer of the Tosefta in the back of our Gemaras, was a living contemporary of Rebbi and one of Rebbi's students (Rebbi being regarded as a Tanna, but at the beginning of the Amoraim). Tosfos sheds a little light on where the Mishnah and this Tosefta come from in Bava Kamma 94b:
ולכך לא פריך לקמן אלא מברייתות דהוה שונה ר' חייא שהיה תלמידו של רבי...לפי שהמשניות היו קודם רבי אלא שרבי סדרם, אבל ברייתות היה שונה ר' חייא מה שהיה מוסיף רבי על המשניות
And that explains why the Gemara only asks later from the Beraisos that R' Chiya taught, who was a Talmid of Rebbi... since the Mishnayos preceded Rebbi, and Rebbi merely arranged them, whereas the Beraisos that R' Chiya learned were what Rebbi added on to the Mishnayos.
So, the Mishnayos, per Tosfos, were merely arranged by Rebbi but hold the strength of an earlier generation. R' Chiya recorded Rebbi's teachings on the Mishnayos and arranged them into the Tosefta (with his student R' Oshaya), as mentioned in Chullin 141.