There is a very sharp break at ch. 40 in tone, style, language, theology, etc., and it is generally accepted that what follows is an addition from the mid-sixth century BC, at a time contemporaneous with King Cyrus of Persia (prior to his conquest of Babylon). The author of this material is usually called Deutero-Isaiah. It should be borne in mind that nowhere does ch. 40-66 claim to have been written by the prophet Isaiah and nowhere does the prophet's name appear (unlike the earlier chapters where the name appears often). It is possible that an anonymous writer (or writers) added his own material to Isaiah or it is possible that this was originally an independent book that somehow accidentally got associated with Isaiah and was copied as part of the book (less likely imho but still possible). It is clear however that this is a separate unit (or units, if indeed ch. 40-66 is composite) from what precedes it on contextual, stylistic, and theological grounds. Concerning the first line of evidence, S. R. Driver said it best:
"The prophet never abandons his own historical position, but speaks from it. So Jeremiah and Ezekiel, for instance, predict first the exile, then the restoration; both are contemplated by them as still future; both are viewed from the period in which they themselves live. In the present prophecy [of Deutero-Isaiah] there is no prediction of exile; the exile is not announced as something still future; it is presupposed, and only the release from it is predicted. By analogy, therefore, the author will have lived in the situation which he thus presupposes, and to which he continually alludes" (Introduction to the Literature of the Old Testament, p. 237).
For instance, the author offers comfort and consolation to his audience (Isaiah 40:1) -- who constitute not the readership of Isaiah addressed in earlier chapters (who in ch. 1-39 were made up of those experiencing Assyrian domination) but plainly the Jewish exiles in Babylon who would experience release by Cyrus. Cyrus is referred to not as a figure from the distant future but a contemporary who has been already designated by Yahweh as his "anointed". The desolation of Jerusalem and the exile of the Jews are not prophesied as these have already happened (cf. Isaiah 42:22, 25, 43:26-28, 44:26-28, 47:6, 52:5, 58:12, 61:4, 63:18, etc.), the prophet instead foretells the end of captivity and the rebuilding of Jerusalem. Thus the author is able to rhetorically appeal to the past in order to make his point to his readership: "Who handed Jacob over to become loot and Israel to the plunderers? Was it not Yahweh against whom we have sinned? For they would not follow his ways and did not obey his law, so he poured on them his burning anger, the violence of war. It enveloped them in flames yet they did not understand". It is not just a matter of assuming a future situation but that the author was engaging with his readers to reflect on past events and the present circumstance of ruined cities and desolated Temple.
The literary style also sharply diverges at ch. 40 and there are many expressions that in combination distinguish the two authors; in Deutero-Isaiah one encounters such frequent key words and phrases as "choose"/"chosen", "praise", "to shoot/spring forth", "break out into singing", "pleasure", "your-fem. sons", "rejoice", "I am" language, DN + participial epithet, "isles"/"coasts", "nought", "arm of Yahweh", etc. but these are absent in the rest of Isaiah, along with such techniques as word duplication, repeating the same word across adjacent clauses or verses, and so forth. In contrast, Isaiah uses such distinctive features as nature imagery, structured antitheses, and other phrases and expressions that do not occur in Deutero-Isaiah. To quote from Driver again: "Isaiah shows strongly marked individualities of style: he is fond of particular images and phrases, many of which are used by no other writer of the OT. Now, in the chapters which contain evident allusions to the age of Isaiah himself, these expressions occur repeatedly; in the chapters which are without such allusions, and which thus authorize prima facie the inference that they belong to a different age, they are absent, and new images and phrases appear instead. This coincidence cannot be accidental. The subject of ch. 40-66 is not so different from that of Isaiah's prophecies (e.g.) against the Assyrians, as to necessitate a new phraseology and rhetorical form: the differences can only be reasonably explained by the supposition of a change of author. Isaiah in his earliest, as in his latest prophecies (ch. 29-33; ch. 37, written when he must have been at least sixty years of age), uses the same style, and shows a preference for the same figures; and the change of subject in ch. 40-66 is not sufficiently great to account for the marked differences which here show themselves. and which indeed often relate to points, such as the form and construction of sentences, which stand in no appreciable relation to the subject treated" (p. 238).
The theology is another key piece of evidence. Deutero-Isaiah is the earliest work that makes the leap to full monotheism, as opposed to the henotheism and monolatry of earlier writers. Deutero-Isaiah's characterization of God as having "no other", who is alone, who is the first and the last, etc., is unique in the OT.
The Society makes a fallacious argument in appealing to the Dead Sea Scrolls. The only thing the Qumran scrolls of Isaiah show is that Deutero-Isaiah was part of Isaiah in the late second century BC, which is not anything anyone would have doubted. What would "refute" the claim that the text is composite is a copy of Isaiah that dates to the seventh century BC that already has ch. 40-66. The Dead Sea Scrolls -- while a millennium older than the Masoretic Text -- are still not old enough to prove that ch. 40-66 formed part of the original text, as they date some 500 years or so too late. One could argue on similar grounds for the authenticity of the Johannine Comma in 1 John (or some other spurious addition to the NT) by appealing to a text some 500 or 600 years later. Of course a late copyist would have no idea of the early transmission history of the book he was copying.
