What is the difference between galli and metragyrtai?

We do not have to assume that there is a substantial difference in the persons of 'one of the galloi' or 'one of the metragyrtai'. Meaning it is not a given that 'galloi did that and were this, but in contrast metragyrtai are very different in doing things'. But we can easily assume that two words describe the exact same pool of persons, only with different contextual meanings. It seems to almost tell us more about those who describe them then about those described. Both are used in an unflattering way by outsiders to describe what they do not entirely 'like'.

We need first to consider who and what the metragyrtai were. The term was derived from two Greek words, Μήτηρ, Mother, and ἀγύρτης, a collector, taken from the verb ἀγείρειν, meaning to gather or collect; a metragyrtes was, then, "one who gathers for the Mother." The word denoted a priest of Meter who went around begging alms for the cult (and probably also for himself). The earliest citation of the word occurs in the fourth-century comic poet Antiphanes, where the metragyrtes is clearly an object of ridicule. Aristotle defined the metragyrtes as ἂτιμος, someone dishonorable, contrasting a metragyrtes with a dadouchos, a torch bearer, an honor­ able religious office. Athenaios, describing Dionysios of Syracuse, commented that he spent his last days as a metragyrtes, a mark of how low the former Sicilian tyrant had sunk.
[…]
For this reason, the current weight of scholarly opinion is that the metragyrtes story records an active tradition of resistance within Athens to the cult of the Phrygian Mother Goddess.
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At this point it is difficult to determine whether the legend of the metragyrtes is essentially a fiction, created to explain the prominence of the Metröon in Athens, or whether it was prompted by the actual execution of metragyrtes on the charge of profaning the Eleusinian Mysteries, preserved by collective memory in the more familiar form of a resistance myth. The story, however, suggests as a historical context, not the cult's foundation, but rather a reaction to the cult of Meter in the late fifth or the early fourth century B.C., during which time a metragyrtes had become Symbolic of the disgraceful Oriental barbarian. […]
At first glance the central location of the Metröon, its: splendid cult statue, and the frequency of votive offerings in the vicinity do not seem to support this negative judgment. Yet the metragyrtes legend is one of several factors; creating the impression that by the late fifth century, the cult of Meter had acquired a distinctively negative tone.
[…] …we should note that one of the most damning charges laid against the metragyrtes in the foundation legend of the Athenian Metröon is that he had come to initiate the women of Athens into the mysteries of Meter. One can read this challenge to traditional authority as part of the mythic pattern of the story, but it may also be that the tradition preserved some truth, that women did find the cult of a mother goddess more appealing… […] Although almost always a term of scorn in literature, the word metragytes never appears in any epigraphical document dealing with the cult of Meter, and there is no indication that the agermos consisted of begging priests passing the hat.

In both situations, the Galli were met with respect, although Livy does comment on their strange appearance and fanatical songs. It is noteworthy, though, that in contexts where the Galli were engaged in serious diplomatic activity, their appearance and sexual status are not used to degrade them.

In these epigrams, however, we see an individual whose appearance, actions, and; sexual status mark him as a deviant. To an extent, this is not surprising; the Gallos was a descendant of the metragyrtes of fourth-century sources, as is clear in one epigram where the word metragyrtes is substituted for Gallos and the metragyrtes of earlier Greek literature was clearly a figure of disdain, Yet the appearance of thel metragyrtes is rarely a source of comment, and his sexuality is never mentioned. In contrast, the sexual status of the Galli in the epigrams is clearly a point of emphasis. They were castrated (6.234) and effeminate (6.217). Moreover they had distinctly feminine appearance and personalities: they had long loose hairs (6.217, 219,220,234), sometimes perfumed (6.234), and they wore women's clothing (6.219). In the course of their rituals, they shrieked (6.219,234), waved their hair wildly (6.21S, 219, 220) and banged on various noisy instruments (6.217,218, 220J 237). This same image occurs in a passage preserved by Hephaestion, ascribed by him to "one of the more recent [i.e., Hellenistic] poets"; here it is the feminine form "Gallai" that refers to the wandering priests of Meter, with their erratic behavior and; use of raucous music:

The roaming thyrsus-loving Gallai of the Mountain Mother clash their instruments and bronze castanets

Was the picture of the Galli drawn by the Greek poets accurate? Several of the activities described in the epigrams were not new to the Meter cult; the participants in the rituals illustrated on the Ferrara krater also toss their hair about; while striking tambourines and castanets. The implication here, however, is that such behavior is indicative of deviant sexuality and effeminacy. The Galli's activities in honor of their goddess served as a form of caricature at best and of degradation at worst.

We have no information suggesting that castration and effeminacy were typical of the Meter cult. The Galli never appear in any cult regulation or decree, and their ac­tivities seem to have been limited to Asia Minor (no Galli are attested in the Piraeus material, for example.) The Hellenistic epigrams appear to exaggerate the characteristics of the Galli to create an artificial literary image, that of a despised group of castrati (the feminine form Gallai anticipates Catullus 63) and foreigners whose eccen­tric appearance and behavior put them beyond the pale of respectable society.

These references to Meter and her priests in Hellenistic literature, although brief and fragmentary, exhibit several trends. We see no personal accounts of religious rit­uals observed or emotional involvement experienced, as was the case in the descrip­tions of Pindar and Euripides. Instead, Meter has become a figure of lighthearted mockery, and her legendary background, her priests, and her rituals are flippanty dismissed.

_{— Lynn E Roller: "In Search of God the Mother: The Cult of Anatolian Cybele", University of California Press: Berkeley, Los Angeles, 1999.}

That is one very plausible take on this. But it cannot be the final word on this. There are multiple ways to explain this, none of them completely conclusive. But if the crux is to whether metragyrtai also mutilated their bodies:

The Galli's adoration of Cybele is a compensation and a regression to the undisturbed relation between a mother and a son, and the castration secures for ever this religious illusion, because no genital sexuality is threatening. Weigert-Vowinkel writes (1938: 372):

Thus the follower of Attis, in the thralls of the Great Mother, renounces his own individuality, he returns to the lap of the mother, who is reconciled by his self-punishment, he returns to a plant-like feminine-childish dependence on her. As the castrated youth, he resembles the feminine deity; in his conversion into a tree, he is the symbolic penis of mother earth, which draws its strength from her alone. The surviving Cybele worshipper withdrew from the masculine rivalry. From that time on, he led the childishly protected life of the Metragyrtes, the begging monks.

_{— J Peter Sodergard: "The Ritualized Bodies of Cybele's Galli and the Methodological Problem of the Plurality of Explanations", Scripta Instituti Donneriani Aboensis, 1993. (PDF)}

For how these people were usually treated in the more prolific centres of writing, but not in others (meaning Athenians and Romans appearing as having had a more 'conservative' view than eg Ionian Greeks) , because of their status, concerning sexuality, (third) gender, religious standing, deviance, body modification and often just presumed behaviour, cf

– Will Roscoe: "Precursors of Islamic Homosexualities", in: Stepehn O. Murray & Will Roscoe (Eds): "Islamic Homosexualities. Culture, History and Lietrature", New York University Press: New York, London, 1997. (p55–86).