Where literary criticism is concerned, what are the major differences between Russian Formalism and New Criticism?
Russian Formalism was probably the first school in literary theory that looked at poetry and literature in general as true linguistic phenomena. One of its representatives, Viktor Shklovsky coined the term defamiliarization (ostranenie in Russian; "Verfremdung" in German), which refers to the intentional deviation from practical, everyday language in poetry and literature. In a much quoted passage from his article "Art as Technique" (1917), Shklovsky wrote,
The purpose of art is to impart the sensation of things as they are perceived and not as they are known. The technique of art is to make objects "unfamiliar," to make forms difficult, to increase the difficulty and length of perception because the process of perception is an aesthetic end in itself and must be prolonged. (...) Art removes objects from the automatism of perception in several ways.
Russian Formalism focused on how poetic language achieved this; its research focuses on poetic language as such and not on individual works of literature. They were not so much interested in analyses that uncovered the meaning of literary texts but in what mechanisms constitute the literariness (German: Literarizität) of texts. This goal was clearly expressed in the name that a number of linguists and literary critics chose for circle in St. Petersburg: OPOJAZ, which was a Russian abbreviation for "Society for the Study of Poetic Language".
Unlike Russian Formalism, New Criticism was not a school in the strict sense of the term, since its principles were only defined in later writings in defence of New Criticism, e.g. Cleanth Brooks's article "The New Criticism" in 1979. There were earlier manifestos, such as The New Criticism by John Crow Ransom in 1941, which already criticised other representatives of New Criticism.
New Criticism emphasised the autonomy of works of literature, which led to criticism that focused on literature's inherent features (or what Germans neatly call werkimmanente Interpretation). Also important was the inherent ambiguity of literature, especially after the publication of William Empson's Seven Types of Ambiguity (1930). The intense focus on the literary text as such is something that New Criticism had in common with Russian Formalism. The New Critics practised close reading, which looked for nuances in meaning and linguistic effects in texts. (I have never seen the term "close reading" in discussions about Russian Formalism.) The New Critics looked for ambiguities; they preferred texts that were complex and may contain ambiguities. This focus is narrower than the focus on the mechanisms of poetic language in Russian Formalism. In addition to ambiguity, New Criticism was also interested in how literary texts achieved unity in spite of ambiguities, paradoxes etc. This became visible in titles such as "The Theme of X in ...". This also shows an interests of works of literature as such, instead of putting the study of literature in the service of finding the mechanisms of poetic language, as the Russian Formalists had done.