Short Answer
The difference between atheistic scientific hypothesis and theory and scientifically oriented faith-based approaches to constructing reality is subtle. Both rely on explanation, but to understand what separates them, one must understand the difference in the type of explanation. Science relies heavily on explication whereas religion relies heavily on narration. Thus, scientific hypotheses should presume the minimum to show what exists from rational and empirical experience, not use stories to assume what exists and attempt to find hypotheses to prop up the presumptions of the story.
Long Answer
Let's start with Carl Sagan, a very popular scientist. In his book "The Varieties of Scientific Experience, Chapter 6, the God Hypothesis, begins recounting that science emerged from religion in a discussion about natural theology. That is to say, that one's epistemology can emerge not solely from divine revelation, but also from religiously inspired reason. In what might be seen as an irony, according to Sagan on p. 148:
The Romans called the Christians atheists. Why? Well, the Christians had a god of sorts, but it wasn't a real god... [a]nd that general sense that an athiest is anybody who doesn't believe exactly as I do prevails in our own time.
Natural theology, which continued the journey started by the pre-Socratics of relying more and more on reason and less and less on revelation and authority for the establishment of truth, means that one's ontology is logically consistent because it relies more heavily on a theory of truth which values coherence. And here then is the guiding demarcation between two types of explanation, that of explication and that of narration. Let's consult Blackwell's Companion to the Philosophy of Science on explanation.
According to the editor in his article on explanation starting on page 127, there are several forms of explanation, and none is fully explanatory of explanation. (Think about it.) Two poles of explanation, then, can be witnessed as one that satisfies formal logical criteria, such as the Deductive-Nominological (D-N) model, and others rely heavily on pragmatic, intuitive efficacy. In fact, Newton-Smith addresses Bas van Fraassen's pragmatic class of explanation:
Van Fraassen whose work has been crucially important in drawing attention to the pragmatic aspects of explanation has gone further in advocating a purely pragmatic theory.
On one extreme, the D-N model requires explanation to walk the path of certainty of deductive logic (no small feet) a particular type formal system, and on the other hand, pragmatic explanation requires nothing more than utility and intuitional appeal that is inherent in informal logic (with all of its blemishes) and is native to storytelling.
So, at the beginning, there was religion based on the myths generated by divine revelation and mystical experience, then natural theology emerged emphasizing reason, and finally, reason under science began producing thinkers who used reason to reject revelation and mysticism as pathways of truth. This is where the modern-day athiest, friendly or militant, exists.
Thus we arrive at your question as I understand it. In a discussion between an atheist and a theist (note how a single space alters the semantics!), where both men accept naturalism, but one rejects supernaturalism, we have a mutual recognition that God is invoked regarding gaps in scientific theory albeit with a slightly different meaning. As an athiest, my experience has been god-of-the-gaps tends to be used as a pejorative precisely because it is seen as a particular species of fallacy related to appeal to ignorance. From WP:
The term "God of the gaps" is sometimes used in describing the incremental retreat of religious explanations of physical phenomena in the face of increasingly comprehensive scientific explanations for those phenomena.
Thus, what separates scientific hypothesis from the pseudo- or pre-scientific hypothesis of theology (revelatory or natural), is that one must fundamentally reject the axiom that God exists as to avoid teleological impulses. This means that scientific hypothesis and theory uses evidence to reach a conclusion, whereas theology uses the conclusion to reach the hypothesis. Simply put, when one presumes a god, one must align one's theory to maintain the presumption. An athiest, such as Dawkins in his The God Delusion, documents how poorly reason functions when it is tethered to presumptions of religious doctrine from thousands of years ago.
The end effect? Scientists give supremacy to explication, whereas theologians to narration. For an athiest, science must prove God, but for a theist, science at best is an instrument to validate God's story. Hence natural theology arguably reaching a pinnacle with the Deists of the Enlightenment. Of course, Jay Gould pushed NOMA, as a way to keep the peace between science and religion.
Appendix
Because "God of the gaps" takes God not as a hypothesis to explain the gaps, which is then to be tested by other means, but as the conclusion from the existence of gaps. Absence of evidence (gaps) is taken as evidence of absence (of a scientific explanation). Such a leap is only justified if no good reason exists for the absence of evidence (which is not the case, science has limited time and resources), and/or alternative explanations are plausibly ruled out independently (which is also not the case). This is not at all how scientific hypotheses are handled. – Conifold
