The relation between the phrases 'positivism' and 'scientism' is ambiguous. Some use 'scientism' as a synonym for 'positivism' (see Leszek Kolakowski's uses in The Alienation of Reason ; e.g., p. 160 and pp. 177-8). Others take positivism to be a proper subset of scientism (my preferred use). Still others take the two groups to be disjoint.
An example of this latter use can be found in D. G. Charlton's admirable book Positivist Thought in France during the Second Empire, 1852-1870. (Oxford University Press, 1959) Charlton reserves 'positivism' for the view of those positivists who remained true to the anti-metaphysical tenets of that philosophy and who did not pretend to draw ethical results from science (or from reason). 'Scientism' then refers (for Charlton) to distortions of positivism that erect ethical or metaphysical doctrines (e.g., materialism) on a putatively scientific basis (Charlton 1959, p. 2; cf. p. 224). Charlton's use of 'scientism' is clarified by this line about Claude Bernard: 'He rejects the claim of scientism that science alone can offer a new ethic of life and reorder civilization'. (Charlton 1959, p. 81)
Chapters 3-7 of Charlton's book distinguish between, on the one hand, those authors who stayed true to positivist epistemology by remaining agnostic about metaphysics and ethics (and much else) and, on the other hand, those who betrayed positivism by laying claim to knowledge beyond what was verifiable, thereby lapsing into scientism. By Charlton's lights, only Bernard and Émile Littré were true positivists while Comte, himself, Ernest Renan, and Hippolyte Taine had unwittingly forsaken positivism for scientism. Here's an interesting remark that Charlton makes about Taine:
He illustrates with especial clarity one of the most significant distortions of positivism in the mid-nineteenth century, a distortion arising from the intermingling of German idealism and Anglo-French positivism. The addition of Hegel and positivism produces scientism: this is the equation demonstrated in the philosophies of Taine and Renan alike. (Charlton, Positivist Thought in France during the Second Empire [Oxford University Press, 1959], p. 154)I quote this intriguing passage but, won't delve into it here.
Charlton's use of 'scientism' connects with what I said in the final paragraph of post 11. Indeed, much of Charlton's book is devoted to 'catching out' supposed positivists in moments of betrayal.
In short, for Charlton, since science is silent on metaphysics, ethics, etc., positivism urges silence on such matters.* 'Whereof one cannot speak, thereof one must be silent' (though I don't mean to say Wittgenstein was a positivist). For Charlton, scientism won't shut up about the 'whereof'.
*The nature of this urging is unclear. It's normative, so, by positivist standards, it isn't based on scientific method, which itself consists of normative claims....Though perhaps at that point they rely only on hypothetical imperatives.