As you correctly note, Russian Formalism and New Criticism have many similarities, as both are part of the early twentieth-century formalist movement. In both schools of thought, the text itself is paramount and is studied independently of its context or author's intent. Literary language is thought of as distinct from ordinary language, and the form and structure of the text are considered to provide more meaning than the content itself. These schools of criticism are most commonly applied to poetry and encourage very close reading.
However, there are several distinct differences between the two schools of thought. In Russian Formalism, a differentiation is made between form and content, whereas New Criticism does not make such a differentiation, maintaining that texts are unified through their patterns, literary devices, themes, etc. Russian Formalism also affords some importance to the text's language and structure, whereas New Criticism considers a text to be completely self-contained, autonomous of its fabric. Furthermore, in Russian Formalism, a process called "dematerialization" was privileged, wherein reality of fact is created through the use of language.