Tough question. Both schools (American/English New Criticism and Russian Formalism) are types of Formalism. Both theories aim at deriving meaning, not from content, but in the structure and form of literature.
Russian Formalism stressed that what a piece of literature "means" cannot be dismissed from how it says it. In other words, the structure/style of literature is not just artifice: it is what makes it meaningful. For example, in mathematical language, 2+2=4, not because of an author's intent (intentional fallacy -New Critics) or a reader's subjective response (affective fallacy - N.C.) or because of historical context. 2+2=4 means what it means because of how it is written. Both New Criticism and Russian Formalism sought to make literary criticism more objective/scientific in this way: to distinguish, formalistically, literary language from ordinary language.
Both schools were criticized for ignoring the historical and sociological components that influenced both the writing and reader reception of literature: not just because it seemed politically irresponsible but because subjective interpretation is part of any active reading, even one by a critic him/herself.
Here's where I think the two schools differ:
Russian Formalism focused on defamiliarization, a making the world seem strange or new. This was what made literature literary. For writing to have literariness, it had to meet certain formalistic criteria: i.e., using language in such a way that its meaning was its form: not tied to, or created by history or outside sources. To be innovative, literature must say things about the world in a new, and necessarily (at least initially) strange way. New Critics saw form, not so much as an opportunity for innovation but as the place for restraint, where criticism must become more scientific and objective. As you might guess, from this difference, Russian Formalist critics tended to be more anti-establishment and New Critics (especially southern American) were reactionary.
Main difference: Russian Formalists focused on form as innovation and American New Critics focused on form as tradition.