Presumably, R' Hoffman is referring to two mourning practices that the Talmud Bavli lists as required but that Tosafot discuss being no longer practiced in their time and place: covering one's face and overturning one's bed.
The Talmud says in Mo'ed Katan 15a:
אבל חייב בעטיפת הראש מדקאמר ליה רחמנא ליחזקאל (יחזקאל כד, יז) ולא תעטה על שפם מכלל דכולי עלמא מיחייבי
A mourner is obligated to wrap his head as a sign of mourning, covering his head and face. This is derived from the fact that the Merciful One says to Ezekiel, while he is in mourning: “And cover not your upper lip” (Ezekiel 24:17). God commands Ezekiel not to display outward signs of mourning, which proves by inference that everyone else is obligated to wrap their heads in this manner.
A little further down, it says:
אבל חייב בכפיית המטה דתני בר קפרא דמות דיוקני נתתי בהן ובעונותיהם הפכתיה כפו מטותיהן עליה
A mourner is obligated to overturn his bed, so that he sleeps on the underside of it, as bar Kappara taught a baraita that states: God stated: I have placed the likeness of My image [deyokan] within humans, as they were created in My image, and owing to their sins I have overturned it, as when this person died the Divine image in him was removed. Therefore, you must also overturn your beds on account of this.
In the midst of a discussion of other restrictions on mourners on 21a1 (s.v. "אלו דברים שאבל אסור בהן"), Tosafot discuss why these two requirements are no longer observed:
ומה שאין נוהגין עתה לעשות עטיפת הראש וכפיית המטה סמכינן אהא דאמרינן בירושלמי אכסנאי אינו חייב בכפיית המטה דלא לימרון חרש הוא פי' מכשף וכמו כן אנו בין הנכרים ויש בינינו עבדים ושפחות ... ועוד יש ליתן טעם בעטיפת הראש שלא היה מביא כי אם לידי שחוק ענין עטיפת ישמעאלים.
Why do people not wrap the head and invert the bed nowadays? They rely on the Yerushalmi, which says that a guest is not obligated in inverting the bed, lest they say [that he is a] Cheresh, i.e. witch; Similarly, we live among Nochrim, and there are male and female slaves among us [who could accuse us of witchcraft]. ... If we wrapped the head, this would lead to laughter, for wrapping is the way Yishmaelim wrap [and no one does so nowadays].
In the case of inverting the bed, Tosafot say explicitly that one of the possible reasons for no longer doing this is to prevent a reaction from gentiles. It seems that this practices started getting dropped by the time the Jerusalem Talmud was written.
Regarding covering the face, Tosafot mention a mirthful reaction without specifying the source, but at least one later authority narrows this concern as well to gentile reactions. In Yore Deya 386:1, the Rema approvingly cites the practice of mourners not covering their faces, and the Shach, presumably based on this passage in Tosafot, explains:
מפני שמביא לידי שחוק גדול מן העובדי כוכבים ועבדים ושפחות שבינינו
Since it causes great mirth from the gentiles and male and female servants among us.3
Given that the Shulchan Aruch there does prescribe covering one's face, and the Rema is the one who approves of not doing so, I would surmise that this practice was dropped first in Ashkenazi communities, by the time that Tosafot wrote about it, and only spread to Sefaradi communities (if it did indeed) after the Shulchan Aruch was written.
These sources don't indicate on whose authority the practices change. As mevaqesh noted in a comment on the question, "The majority of changes in minhag receive at most ex post facto justification."
1. Thanks to DoubleAA for pointing out this source in a comment on the question.
2. Thanks to rosends for pointing out this source in a comment on the question.
3. My translation.