I heard this said by Dr. Jordan B. Peterson. I doubt that it's correct. Are there mitzvot more important than others? I know that all 613 are contained in them but that doesn't make the 10 more important. I remember learning that in certain circumstances, reading the 10 commandments aloud in shul is prohibited. Do you know of the source for this?
In regard to your second question, they Ten Commandments were included in the Temple service, more than 2,000 years ago, and recited before the Shema, then they were removed because heretics said they were the only valid commandments [Berakhot 12a; Mishnah 5:1 in Tamid 32b]. All attempts to reinstate them failed.
But we are allowed to recite the Ten Commandments outside of services. Jews in Fostat, near Cairo, Egypt, in the 10th-12th centuries, regularly read the Ten Commandments after services [Jacob Mann, The Jews in Egypt and in Palestine under the Fatimid caliphs I, p 221]. The Baal Ha-Turim, from 14th-century Spain, said that people should say them privately before services. Rabbi Yosef Karo, from 16th-century Israel [Shulḥan Arukh, Orach Ḥayyim 1:5] confirms in his Code of Jewish Law that the ban only applies to public recitation during services. Rabbi Moshe Isserles, from 16th-century Poland, said that one may recite them privately at any time, but not in public. They are included for that purpose in many modern prayerbooks, such as Artscroll.
Judaism does not rank the commandments. The Mishnah is very explicit on that point:
וֶהֱוֵי זָהִיר בְּמִצְוָה קַלָּה כְבַחֲמוּרָה, שֶׁאֵין אַתָּה יוֹדֵעַ מַתַּן שְׂכָרָן שֶׁל מִצְוֹת. רַבִּי אוֹמֵר Rabbi [Yehudah haNasi] said: Be as scrupulous in observing a [seemingly] minor commandment as a [seemingly] major commandment, because you do not know the value of each commandment. [Pirkei Avot 2:1]
Note that this quote says that there is a value to each mitzvah. We just don't know what it is. We do not know enough to rank mitzvot. A specific mitzvah may be worth dozens of other mitzvot. Maimonides writes:
Only the Master of Opinions [that is, God] knows how the comparison between sins and merits is made. [Rambam, Mishneh Torah, Hilkhot Teshuvah 3:1-2]
Berachos 12A when discussing if we should recite the Ten Commandments every day, says;
אָמַר רַב יְהוּדָה אָמַר שְׁמוּאֵל: אַף בִּגְבוּלִין בִּקְּשׁוּ לִקְרוֹת כֵּן, אֶלָּא שֶׁכְּבָר בִּטְּלוּם מִפְּנֵי תַּרְעוֹמֶת הַמִּינִין.
Rav Yehuda said that Shmuel said: Even in the outlying areas, outside the Temple, they sought to recite the Ten Commandments in this manner every day, as they are the basis of the Torah (Rambam), but they had already abolished recitation of the Ten Commandments due to the grievance of the heretics, who argued that the entire Torah, with the exception of the Ten Commandments, did not emanate from God (Jerusalem Talmud). If the Ten Commandments were recited daily, that would lend credence to their claim, so their recitation was expunged from the daily prayers. (Sefaria)
You ask, "Are there mitzvot more important than others?"
The other answers have addressed your question from the standpoint of, are some mitzvot more important to perform than others. But the question can also be answered from the perspective of, are some mitzvot more important to not be broken.
The concept of pikuach nefesh addresses this point. It states that of the 613 mitzvot, all but 3 may be broken in order to save a human life. The 3 mitzvot that cannot be broken include
- defaming G-d's name
- forbidden sexual relations
Hence, these 3 mitzvot are more important not to break than the other 610.