The Mechaber says in the context of Hilchos Chazaras HaShas that after the name of Hashem is said in a bracha the congregation should say out loud the words, “Baruch Hu U’mivorach Shmo”.
On this the (Gr”A) Vilna Gaon, comments and says that one should not say “Baruch Hu U’mivorach Shmo”, rather he should remain quiet until the end of the bracha, and then say Amen.
R’ Soloveitchik quoted the Sefer Nefesh HaChaim as a possible reason for why “Baruch Hu U’mivorach Shemo” should not be said.
The Rav said that every time that we mention the name of Hashem in prayers, the reference is entirely made on the characteristics of Hashem, not Hashem “himself” (biatzmuso). This is why we say in kedusha, “Nikadesh es Shimcha…” in Birchos Krias Shma, “Es shem…” and in Kadish, “Yihei Shmei Rabba…” The reason for this is that it is impossible to give praise and to "someone" that we don’t have the slightest recognition of.
By praising the name (shem) of Hashem however, we are recognizing individual characteristics and happenings that we are aware of.
There is one time in the Torah however, when Hashem himself is said to be kadosh. This is in the words, “Vikidashti bisoch Bnei Yisroel”?! The obvious answer to this concern is that in this Pasuk HaKadosh Baruch Hu is talking about himself. Of course HaKadosh Baruch Hu could say this about himself.
This idea can be evoked even more so from the first psalm of the Hallel Prayer. We start the prayer by saying, “Haliluka Halilu es shem Hashem, Yihi shem Hashem Mivorach” We see from this verse that the only time when Hashem can be praised is if it is in the context of his name, NOT “biatzmuso”!
Therefore it is improper to say, “Baruch Hu Umivorach Shemo” because one is not only praising Hashem’s name but Hashem “biatzmuso”.