“Islamicate” vs “Islamic”
Marshall Hodgson lays claim to the invention of the term in the introduction to his history, The Venture of Islam, Volume 1: The Classical Age of Islam (University of Chicago Press, 1975). There are sources which suggest he coined it and used it in essays and other works starting in the 1960s, but Venture does provide a complete explanation of it, part of a lengthy discussion of cultural and geographic terminology:
On the other hand, if the analogy with 'Christendom' is held to, 'Islamdon' does not designate in itself a 'civilization', a specific culture, but only the society that carries that culture. There has been, however, a culture, centred on a lettered tradition, which has been historically distinctive of Islamdom the society, and which has been naturally shared in by both Muslims and non-Muslims who participate at all fully in the society of Islamdom. For this, I have used the adjective 'Islamicate'. I thus restrict the term 'Islam' to the religion of the Muslims, not using that term for the far more general phenomena, the society of Islamdom and its Islamicate cultural traditions.…
At any rate, it is already felt improper, among careful speakers, to refer to some local event as taking place ‘in Islam’, or to a traveller as going ‘to Islam’, as if Islam were a country. The adjective ‘Islamic’, correspondingly, must be restricted to ‘of or pertaining to’ Islam in the proper, the religious, sense, and of this it will be harder to persuade some. When I speak of ‘Islamic literature’ I am referring only to more or less ‘religious’ literature, not to secular wine songs, just as when one speaks of Christian literature one does not refer to all the literature produced in Christendom . When I speak of ‘Islamic art’ I imply some sort of distinction between the architecture of mosques on the one hand, and the miniatures illustrating a medical handbook on the other even though there is admittedly no sharp boundary between. Unfortunately, there seems to be no adjective in use for the excluded sense-‘of or pertaining to’ the society and culture of Islamdom. In the case of Western Christendom we have the convenient adjective ‘Occidental’ (or ‘Western’-though this latter term, especially, is too often misused in a vaguely extended sense). ‘Occidental’ has just the necessary traits that ‘West Christian’ would exclude. I have been driven to invent a term, ‘Islamicate’. It has a double adjectival ending on the analogy of ‘Italianate’, ‘in the Italian style’, which refers not to Italy itself directly, not to just whatever is to be called properly Italian, but to something associated typically with Italian style and with the Italian manner. One speaks of ‘Italianate’ architecture even in England or Turkey. Rather similarly (though I shift the relation a bit), ‘Islamicate’ would refer not directly to the religion, Islam, itself, but to the social and cultural complex historically associated with Islam and the Muslims, both among Muslims themselves and even when found among non-Muslims.
The pattern of such a double adjectival ending, setting the reference at two removes from the point referred to, is sufficiently uncommon to make me hesitate. But there seems no alternative. In some contexts, but only in some, one can refer without ambiguity to the 'Perso-Arabic' tradition to indicate 'of or pertaining to' Islamdom and its culture, for all the lettered traditions of Islamdom have been grounded in the Arabic or th Persian or both. In other cases, one might use a periphrasis involving the terms 'traditions/culture/society of Islamdom'. One cannot, speaking generally, call Swedish 'a Christian language'; and if one were debarred from calling it an 'Occidental' language, one could not say simply that it is 'a language of Christendom', which might in some contexts seem to imply that it was to at least some extent used throughout that extensive realm; but one might say it is 'a language of the culture of Christendom'.
This level of distinction is not commonly needed or even comprehended outside of area studies, and the term is quite rare in non-academic publications. It does not turn up at all in several of the BYU corpora including Google Ngrams, BNC, or Hansard, and its only entry in COCA is from an art journal from 2009. The News on the Web Corpus turns only 22 entries, all since 2010— a new edition of The Venture of Islam was published in 2009. By comparison, ummah variously spelled has not quite 300 results, and Islam has 12,304.
The website for the 2014 Islamicate Studies Symposium, held on the occasion of the 40th anniversary of Hodgson's work, notes that he was the originator of "a number of neologisms and some very careful usages," Islamicate being one of them.
The term islamicate was coined by Marshall Hodgson with a specific connotation, that is to refer to all forms of cultural expressions which show a clear influence of Islamic, Muslim traditions even though they are originated from non-Muslims or outside Muslim communities. The term may coincide or differ from "Islamic" in this respect, that's probably why you have found it used in accociation or in contrast to it. My impression that the term is not commonly and properly understood and its usage may be found mainly in academic papers:
The term Islamicate culture was coined by Marshall Hodgson (d. 1968) in the first volume of his The Venture of Islam (1974). Hodgson invented the term in response to the confusion surrounding such terms as "Islamic," "Islam," and "Muslim" when they are used to describe aspects of society and culture that are found throughout the Muslim world.
Hodgson used the term to describe cultural manifestations arising out of an Arabic and Persian literate tradition, which does not refer directly to the Islamic religion but to the "social and cultural complex historically associated with Islam and the Muslims, both among Muslims themselves and even when found among non-Muslims" (p. 59). For example, Hodgson argued that there are a variety of artistic, architectural, and literary styles indicative of Islamicate culture. No matter where these aesthetic styles are found, they are identifiable as deriving from Islamicate cultural complexes. Thus, if one finds the use of arabesques, calligraphy, or arched doorways anywhere in the world, these forms are identifiable as Islamicate in origin.
In constrast, Hodgson argued that those elements of Islamic society that are not shared by non-Muslims are not indicative of Islamicate culture (for instance, mosque architecture). Due to the overriding influence of Islam on non-Muslims living within Muslim realms, however, Hodgson used the term to demonstrate the importance of Islam as a cultural force that influenced non-Muslim forms of art, literature, and custom.
From (www.encyclopedia.com)