I thought to myself it must be, in general, considered differently in Judaism, but I haven't found written sources for that yet. Is the first considered differently in the Torah than the second?
Judaism is a very law based religion, and just like any society based on laws there are subdivisions that can add context. A child born of a religiously Jewish father and non Jewish mother is not considered Jewish. This is the law on the books for normative Jewish law.
Having said that, there are some subtleties to the situation which have practical ramifications. In many Middle Eastern Jewish communities for example, if there was a Jewish man from the community who married a non Jewish woman the relationship was not initially accepted by the community. However, if the man continued to attend services, and his wife showed devotion to Judaism within the community, then the community would make conversion for her much easier if she was to become pregnant. Because there is an idea that we want to keep the seed of Israel ongoing.
This "seed of Israel" concept has been officially coined for situations in which a child is born to a Jewish father and non Jewish mother. Certain Rabbis have argued in the past that conversions for these children should be made easier, especially if these children are raised practicing Judaism. I will now quote an answer from another question that applies here and provide a link at the end:
Rabbi Tzvi Hirsch Kalischer was the first to coin the phrase: Zera Yisrael
Rabbi Kalischer used this in reference to someone born to a non-Jewish mother but had a jewish father. He referred to them as Zera Kadosh - Holy Lineage. However, Rabbi Kalischer did not use it in a halachic context.
The most vocal and bold Halachist to apply this concept was the Former Sephardic Chief Rabbi of the State of Israel Rav Ben Zion Meir Chai Uziel (שו”ת משפטי עוזיאל, כרך ב יו”ד סי’ נב) who argued that the concept of Zera Yisrael is considered so legitimate that it can be applied to determine lineage. For example, if the non Jew -zera Yisrael father is a Kohen, despite his mother being a gentile, once he converts his "zera Yisrael" kicks in and he is considered a full fledged Kohen and can partake in the many benefits/ prohibitions that apply to Kohanim.
Rabbi Uziel was deeply concerned about the fate of children born to a Jewish father and a non-Jewish mother. Such children, although of Jewish stock (zera Yisrael), are in fact not halakhically Jewish. Children raised in such intermarriages will be lost to the Jewish people entirely. Thus, it is obligatory for rabbis to convert the non-Jewish mother in order to keep the children in the Jewish fold. Rabbi Uziel noted: “And I fear that if we push them [the children] away completely by not accepting their parents for conversion, we shall be brought to judgment and they shall say to us: ‘You did not bring back those who were driven away, and those who were lost you did not seek.’ (Yehezkel 34:4).”
Some (source for quote above as well) have argued that the concept of Zera Yisrael is also used as a justification to streamline the conversion process. Where traditionally a certain amount of time is given in order to bring the prospective convert up to speed in torah knowledge, or acclimation to an observance life, as well as to discern their true motivation for conversion. Some have argued that in the case of a Zera yisrael, we speed up the process and accept them right away as it is a mitzvah to bring that back under The Wings of the Schechina.
However, this concept (if legitimate) is only limited to those that are literal descendants of Jewish lineage.
Great question! Given that your question has orthodox tagged in, I assume you want an approach of Jewish law. In Judaism, matrilineal descent can be the only defining factor of a persons Jewish status (unless geirus), regardless of whether the father is an "observant" Jew or not, the children can never be Jewish unless they convert and until then they do, they are completely as non-Jews as a person born to 2 non-Jewish parents. Matrilineal descent is evidenced by when Ezra returned back from exile he returned and saw many of the Jewish men taking non-Jewish wives, he then cried out and as the verse states:
“Now then, let us make a covenant with our G‑d to expel all these women and those who have been born to them, in accordance with the bidding of the Lord and of all who are concerned over the commandment of our G‑d, and let the Torah be obeyed.” aka those that are born to non-Jewish women are expelled from being considered Jewish.
The second and most compelling explanation is found:
You shall not intermarry with them; you shall not give your daughter to his son, and you shall not take his daughter for your son. For he will turn away your son from following Me, and they will worship the gods of others, and the wrath of G‑d will be kindled against you, and He will quickly destroy you.- Deuteronomy 7:4
Whilst all marriages between Jews and a non-Jew is forbidden and not considered a marriage this special mention of marrying non-Jewish women turning away your son from following G-d means that they are not considered Jewish once they are born to a non-Jewish woman.
In terms of conversion, it is much easier to convert and exposure to Judaism helps inform the beis din that you are serious and know what you are talking about.
I hope I helped and feel free to ask more questions.
Please read this for further reading:
It depends on a number of factors, in Orthodox Judaism yes you would be considered fully non-Jewish. However they'll often give you an easier time if you want to convert.
In Reform Judaism if you were raised Jewish with either of your parents being Jewish than they consider you a Jew.
In Conservative Judaism it's a more complex question that depends on the community.