The primary type of literary criticism that developed among modernists was formalism, or New Criticism. Postmodernist literary theories include post-structuralism and deconstruction.
"New critics" argued that a text contains a meaning that can be understood apart from external factors. Post-structuralists believe that no overarching theories can be used to interpret a text, while deconstructionists search for instances in which a text contradicts itself.
Practitioners of New Criticism believed that a text has only one, right interpretation. Post-structuralists are concerned with how human systems of thought are expressions of power (and look for examples of such in a text). Deconstructionists distrust the ability of language to transmit truth or meaning and see in texts many "truths."
T.S. Eliot and Northrop Frye numbered among the literary critics in the New Criticism tradition. Examples of post-modernist literary theorists are Jacques Derrida and Michel Foucault.
A "New Critic" would examine the formal components of a text, e.g., rhythm or imagery, whereas a post-modernist would seek ways in which a text urges a particular ideology or ways in which a text subverts societal truths.