Why do Ashkenazim say Baruch Shem Kevod Malchuto L'olam Va'Ed by Tefilin Shel Rosh and not by other mitzvot where saying a beracha is questionable?
The Rama says in Siman 25 seif 5 that the Ashkenazi custom is to say a beracha over Tefilin Shel Rosh, as opposed to the Sepharadi custom to only say one on Tefilin shel yad. The Rama adds that it is good to say “Baruch Shem Kevod Malchuto L'olam Va'Ed” after that second beracha.
The Mishna Berurah explains that the reason why it is good to say Baruch Shem Kevod Malchuto L'olam Va'Ed after the second beracha because there is a big argument among the poskim on whether a beracha is needed here, and usually when there is an argument we do not say a beracha. Yet here we do say a beracha because that is the custom. Nevertheless, we still want to satisfy the other opinion that says not to say the beracha, so we say Baruch Shem Kevod Malchuto L'olam Va'Ed, As we know, whenever someone says a blessing in vain they should say Baruch Shem Kevod Malchuto L'olam Va'ed.
Why is it that with regard to other mitzvot such as Hallel and blowing shofar for women, Ashkenazim say it with a beracha but they do not say Baruch Shem Kevod Malchuto L'olam Va'ed?
Besides the Mishna Berurah's reason mentioned earlier, Aruch Hashulhan says that there is a special connection between the verse Baruch Shem Kevod Malchuto L'olam Va'ed and tefilin, because when one is wearing tefilin he accepts the uniqueness and unity of Hashem, just like when one is saying the verse Baruch Shem Kevod Malchuto L'olam Va'ed. Based on this reason we can definitely say that Baruch Shem Kevod Malchuto L'olam Va'ed is more relevant to tefilin and less to Hallel and shofar, That is why the poskim decided it should be said after the second beracha of tefilin, but not after the beracha for Hallel or shofar.
We can also say that we don't want to have any interruption between the beracha and Hallel, or between the beracha and blowing the shofar, as opposed to tefilin where there is no interruption.
Another reason is that since on some festivals, such as Hanukah and Pesach, all agree that we do say a beracha before Hallel, Therefore the poskim did not want to establish for us to say Baruch Shem Kevod Malchuto L'olam Va'ed by every Hallel. It would be confusing to ordain the verse at only those occasions where there is a dispute over whether a beracha is said for Hallel. Whereas by tefilin, there is always an argument so the poskim felt it more appropriate to ordain it for tefilin rather than hallel.
Many poskim hold that we should not say Baruch Shem Kevod Malchuto L'olam Va'ed after the second beracha of tefilin at all, even for Ashkenazim, because we don't want people to think that they can solve the problem of Safek Berachot with simply saying Baruch Shem Kevod Malchuto L'olam Va'ed whenever there is a doubt as to whether a blessing should be said or not.