Russian formalism and New Criticism are very similar. Both purport to be scientific and rigorous in their analyses of texts. Both sever the text from the biography of the author and its social context. Both are primarily interested in studying the words on the page in isolation from other inputs.
New Critics, however, focused more on understanding the content or the overall meaning of a work of literature than the Russian Formalists. The New Critics wanted to brush away biography and history in order to get a clearer understanding of what a work of literature really meant when encountered on its own. They believed that an author's biography, for example, could get in the way of understanding what a text was saying or divert attention from a text to an author.
The Russian Formalists, however, were less interested in the content of a work or its over all meaning and more interested in focusing on the specific techniques literary writers used to make their works "literary." What, specifically, made a literary text different from a newspaper article or a scientific paper? How did literary authors achieve the effect of what the formalists called "defamiliarization," or making language sound stunning or different from the ordinary?
The New Critics were also very interested in literary techniques but were more focused on using them to illuminate the meaning or content of a work. One way to understand this is to see the Russian formalist as wanting to dissect the "frog" of its text into component parts: the bones would go in one pile, the skin in another, and so on: they weren't particularly concerned with keeping the "frog" as a whole. The New Critics would open up the "frog" of the text and study its component parts but want to keep it together, maybe pinned to a board, so that by understanding the parts, you could understand the frog as a whole.