The Rambam writes in Moreh Nevukhim (1:61):
You must beware of sharing the error of those who write amulets. Whatever you hear of them or read in their works, especially in reference to the names which they form by combination, is utterly senseless; they call these combinations names and believe their pronunciation demands sanctification and purification and that by using them they are able to work miracles. Rational people ought not to listen to such men nor in any way believe their assertions.
In other words, amulets: no good. However, the Rambam rules in Mishneh Torah, Hilkhos Shabbos 19:14:
One may go out [on Shabbos where there is no eruv] with a tested amulet. What is a tested amulet? One which has cured three people or which has been made by a person who has cured three people by means of other amulets.
In other words, amulets: good. How can we resolve this apparent contradiction? R. J. David Bleich offers the following three possible resolutions (“Maimonides on the Distinction between Science and Pseudoscience” in Fred Rosner and Samuel Kottek eds., Moses Maimonides: Physician, Scientist, and Philosopher, pp. 253-254 n. 3:Click here to read more
This contradiction might perhaps be resolved by positing that Maimonides accepted the ruling of the sages regarding entering a public thoroughfare on Shabbat while wearing the objects described in the Mishnah, Shabbat 60a, even though he regarded them as devoid of any therapeutic efficacy. Maimonides might have assumed that the sages ruled in this manner because, since the masses accepted their efficacy, albeit erroneously, such items acquired the status of articles of clothing or of ornaments simply by virtue of being customarily worn as a cure or prophylaxis. See Rashi, Shabbat 60a. Alternatively, Maimonides may have regarded the practice as being permitted because, in light of the fact that the items in question “are not carried in the usual manner,” no transgression of a biblical prohibition is involved. See Teshuvot ha-Radbaz, Le-Leshonot ha-Rambam, V, nos. 63 (1,436) and 153 (1,526); and the comments of the interlocutor as reported in Teshuvot Shemesh Zedakah, no. 29. Since no biblical prohibition is entailed, and since the masses were desirous of wearing amulets because of their misplaced beliefs regarding therapeutic properties ascribed to such amulets, the sages did not choose to disturb the practice with an interdiction against wearing them on the Sabbath.
In short: 1. Wearing an amulet is not carrying but wearing clothing, 2. It is not biblically prohibited and the Sages permitted the rabbinic prohibition that would otherwise be involved. However, R. Bleich rejects these two suggestions without explanation and proposes a third.
Those explanations, however, are entirely unlikely. However, another resolution of Maimonides’ conflicting comments does suggest itself. In context, Maimonides’ comments in the Guide occur in the course of a discussion of the various names of God and indicate that only the tetragrammaton is the nomen proprium of the Deity, while all other appellations are simply reflective of divine attributes indicating the relationship of certain actions to Him but are in no way reflective of the divine essence. Moreover, Maimonides insists that all divine attributes are negative in nature and designed to negate the possibility of certain actions or qualities but tell us nothing of the nature of Deity in a positive sense. Maimonides’ critical comments concerning amulets may, then, have been directed only against writers of amulets containing various divine names or various combinations of divine names and their ascription of supernatural properties to those names. Since Maimonides denies that those divine names define the essence of the Deity, he categorically rejected the efficacy of amulets employing such names. Those names neither reflect the essential nature of the Deity nor do they reflect His qualities or attributes in any positive sense. Thus they cannot conceivably be endowed with any mystic power… The amulets described in the Talmud, to which he refers in Hilkhot Shabbat, may well have been of an entirely different nature. Their nature is, of course, unknown to us. But those amulets, when demonstrated to have been efficacious, were accepted by Maimonides and their curative power acknowledged by him.
In other words, amulets with God’s name: bad. Amulets without God’s name and with proven curative powers: good.