The initial set-up is quite different, naturally. The Green Knight comes in uninvited without any introduction or explanation -- the reader is thus in the same boat as members of Arthur's court -- whereas Tolkien gives us some backstory on the Lieutenant of the Tower of Barad-dûr when he comes out in response to the heralds' challenge. The Green Knight arrives alone on a color-coordinated steed that seems an ordinary animal except for its hue, but the poet hints the knight himself might possibly be supernatural ("Half etayn in erde I hope þat he were"). Intriguingly, the similarly color-coordinated fellow who approaches Aragorn & Co. is almost exactly the inverse, i.e., a living man on a possibly supernatural mount:
[O]ut of [the Black Gate] there came an embassy from the Dark Tower. At its head there rode a tall and evil shape, mounted upon a black horse, if horse it was; for it was huge and hideous, and its face was [...] more like a skull than a living head, and in the sockets of its eyes and in its nostrils there burned a flame. The rider was robed all in black, and black was his lofty helm; yet this was no Ringwraith but a living man.The core similarity, of course, is the disrespectful address. In each version, the stranger boldly rides right up to the company and makes a big show of looking them up and down and asking who is in charge. He is very specifically pretending not to be able to discern the leader -- a matter which would be self-evident both from the man's own physical location, bearing, and adornment and from his followers' reactions, since they would doubtless be turning to him or looking his way. Here's Tolkien in LotR:
(LotR 888, paragraph break omitted)
Now halting a few paces before the Captains of the West he looked them up and down and laughed.
'Is there anyone in this rout with authority to treat with me?' he asked. 'Or indeed with wit to understand me? Not thou at least!' he mocked, turning to Aragorn with scorn. 'It needs more to make a king than a piece of Elvish glass, or a rabble such as this. Why, any brigand of the hills can show as good a following!'Clearly, the Mouth of Sauron knows who Aragorn is, since he specifically picks him and his Elvish glass out for the first round of mockery. Here's the Middle English poet's verse (ll. 221-231):
Þis haþel [knight] heldez [proceeds, goes, comes] hym in and þe halle entres,
Driuande [lit: driving] to þe heȝe dece, dut [feared] he no woþe [danger],
Haylsed [greeted] he neuer one, bot heȝe he ouer loked.
Þe fyrst word þat he warp [uttered], 'Wher is', he sayd,
'Þe gouernour of þis gyng [company]? Gladly I wolde
Se þat segg [man, knight] in syȝt, and with hymself speke
To knyȝtez he kest his yȝe,The Green Knight's words are less overtly disrespectful here; he does not call into question Arthur's intellectual capabilities, compare him to a "brigand," or refer to his followers as "this rout" or "a rabble." Or does he? The word "gyng" (l. 225) stands out initially due to its visual resemblance to "gang." Tolkien's notes and glossary translate it as "company" (1st ed. p. 160) and his own translation uses "gathering" (p. 23). Likewise, Borroff goes with "crowd" in her verse translation, both in the 1967 original and a revised version for the Norton Critical Edition (2010).
And reled [rolled] hym vp and doun;
He stemmed [stopped, halted], and con [did] studie [look carefully, lit: study]
Quo walt [possessed] þer most renoun.
Still, Borroff's commentary on these lines in "The Challenge Episode: A Stylistic Interpretation" cites the OED to suggest the word conveys an ambiguously deprecatory sense.
Indeed, the first several definitions or subdefinitions in OED's entry for "ging, n." (1, 2a, 2b, 3a) are consistent with this neutral usage; it can mean (for example) a company of armed men, a great personage's retinue, household, followers, or retainers, or even more generally a gathering of people. 1, 2a, and 2b are attested at various times from 1043 (in Old English) through 1632, while 3a is attested ?c1200–1877. But then we reach definition 3b:
b. depreciative. A crew, a rabble.As we have seen, the Mouth of Sauron refers to the host as "this rabble."
Curiously, the Green Knight seems to have come indoors ("heldez hym in and þe halle entres") on horseback. He does not dismount on entering the feast-area, but instead drives or presses forward to the high dais ("Driuande to þe heȝe dece," rendered by Tolkien as "pressing forward to the dais"). In this regard, the Green Knight seems more overtly disrespectful than the Mouth of Sauron; an emissary summoned forth to answer a challenge might well ride up to the enemy awaiting him, but an unexpected visitor dropping in at Christmas revels "in halle" (l. 101) should surely, at the very least, approach the dais on foot.
The Green Knight's insolence devolves into increasingly open mockery (ll. 280-86, 309-15) and then to loud laughter (l. 316). Arthur initially identifies himself and graciously invites him to join in the feast and let them know his business after, but the Green Knight declines. He's not there for a fight, of course, because "Hit arn aboute on þis bench bot berdlez chylder" (l. 280). Instead, he challenges them to a beheading game. When his startling offer is met with stunned silence, the Green Knight throws off all restraint (ll. 309-22):
'What, is þis Arthures hous,' quoþ þe haþel þenne,So Aragorn, unlike Arthur, passes the test insofar as keeping his cool under open mockery and laughter: "Aragorn said naught in answer, but he took the other's eye and held it, and for a moment they strove thus," until the challenger quails (LotR 889).
'Þat al þe rous [fame, talk] rennes of þurȝ ryalmes so mony?
Where is now your sourquydrye [pride] and your conquestes,
Your gryndellayk [fierceness] and your greme [wrath], and your grete wordes?
Now is þe reuel and þe renoun of þe Rounde Table
Ouerwalt wyth a worde of on wyȝes speche,
For al dares for drede withoute dynt schewed!'
Wyth þis he laȝes so loude þat þe lorde greued;
Þe blod schot for scham into his schyre face
He wex as wroth as wynde,
So did alle þat þer were.
Þe kyng as kene bi kynde
Þen stod þat stif mon nere,
The similarities in set-up perhaps reflect that, in each case, the emissary seeks to undermine, to provoke, to throw the good guys off their game, and ultimately to set a trap for them.
Coda - Miscellaneous Details
The Green Knight issues his challenge on horseback, since immediately afterward "Þe renk on his rouncé hym ruched in his sadel" to look around at the company (l. 303). Moreover, once the challenge has been accepted, "Lyȝtly lepez he hym to, and laȝt at his honde" (l. 328). So he's kinda doubling down on the not-dismounting thing until he gets what he wants.
The scene in LotR does not include a similar challenge/exchange. The Mouth of Sauron is answering the heralds' challenge: "Come forth! [...] Let the Lord of the Black Land come forth! Justice shall be done upon him. For wrongfully he has made war upon Gondor and wrested its lands. Therefore the King of Gondor demands that he should atone for his evils, and depart then for ever. Come forth!" (LotR 887). But this is a fundamentally different kind of challenge than the Green Knight's proffered exchange of one beheading for another.
The Mouth of Sauron does offers the company an exchange, but it, too, is fundamentally different from that offered by the Green Knight, since the terms are wildly unequal on their face (rather than like for like): He invites total submission and capitulation in return for the non-torture of one hobbit.
Note on Etymology:
From the OED's etymological notes on "gang, n.":
Sense 8 probably developed primarily from the conception of a group of people going about together, whereas senses 9 and 10 were probably additionally influenced by sense 7, as denoting a group or set (of people or animals) having characteristics in common. Compare earlier ging n.1 It is uncertain whether there was any influence from early Scandinavian uses in compounds, or whether these simply show a parallel development; compare Old Icelandic þjófa-gangr group of thieves, gaura-gangr group of ruffians, and also drauga-gangr group of ghosts, músa-gangr group of mice. (Dutch gang and German Gang denoting a group of criminals show borrowings < English.)
In turn, the outline for "† ging, n.1" provides:
Origin: Of uncertain origin. Either (i) a variant or alteration of another lexical item. Or (ii) a borrowing from early Scandinavian. Etymon: i-geng n.
Etymology: Either (i) aphetic < i-geng n., or (ii) < early Scandinavian (compare Old Icelandic ...
1. A company of armed men, a troop, army, host.
a. A retinue (of a great personage); a household, a body of retainers or followers.
b. In plural. A person's followers or people. Also: people in general.
a. gen. A gathering of people, a company; a band, a gang; a set. Also figurative.
b. depreciative. A crew, a rabble.
c. spec. The crew of a ship or boat. Cf. gang n.
4. In Old Testament usage: the Gentile nations collectively; heathen peoples.
NOTE: The citations are rather rough - I'll have to go back and clean them up at some point.
Armitage, Simon. Sir Gawain and the Green Knight: A New Verse Translation. W.W. Norton, 2008.
Borroff, Marie. Sir Gawain and the Green Knight: A New Verse Translation. W.W. Norton, 1967.
---. “The Challenge Episode: A Stylistic Interpretation.” Sir Gawain and the Green Knight: An Authoritative Translation, Contexts, Criticism, edited by Marie Borroff and Laura L. Howes, 1st ed, W.W. Norton, 2010, pp. 93–104.
---. “The Translated Text of Sir Gawain and the Green Knight.” Sir Gawain and the Green Knight: An Authoritative Translation, Contexts, Criticism, edited by Marie Borroff and Laura L. Howes, 1st ed, W.W. Norton, 2010, pp. 1–64.
“gang, n.” Oxford English Dictionary Online, Third Edition, Mar. 2013, http://www.oed.com.i.ezproxy.nypl.org/view/Entry/76566.
“ging, n.1” Oxford English Dictionary Online, Third Edition, June 2017, http://www.oed.com.i.ezproxy.nypl.org/viewdictionaryentry/Entry/78368.
"Sir Gawayn and Þe Grene Knyȝt." Sir Gawain and the Green Knight, edited by J. R. R. Tolkien and E. V. Gordon, Clarendon Press, 1949. (https://quod.lib.umich.edu/c/cme/Gawain/1:1?rgn=div1;view=fulltext)
---. Sir Gawain and the Green Knight, Edited by J.R.R. Tolkien and E.V. Gordon, edited by Norman Davis, Norman, editor, 2nd ed., Clarendon Press, 1968.
Tolkien, J. R. R. The Lord of the Rings. 50th Anniversary One-Volume Edition, HarperCollinsPublishers, 2005.
Tolkien, J. R. R., translator. “Sir Gawain and the Green Knight.” Sir Gawain and the Green Knight, Pearl and Sir Orfeo, edited by Christopher Tolkien, HarperCollinsPublishers, 1975, pp. 17–93.