What are the relationships between formalism and structuralism?

Proponents of Formalism and New Criticism think the analysis of literature should be focused on form. What makes a poem differ from prose? What makes all literature differ from ordinary language? They purposefully ignore things external to the text: things like authorial intent and historical context. This analysis also studies how literature is able to convey universal truths, which, being timeless, are not limited to any biographical or historical context.

These days, Formalist criticism is rarely done in isolation. Word choice in a poem ("she" rather than the historically universal "he") might reflect a growing awareness of gender equality. This introduces a cultural awareness—one that suits a structural analysis more than a formalist one. One of the ways that literary language differs from other language is that it can make the reader look at something (in life) in a new way. This, in formalist lingo, is called defamiliarization: the act of presenting ordinary things in a strange way. This is done through the use of literary (and some times experimental) language. Formalism is a disciplined look at literariness.

One of the main similarities between formalism and structuralism is the focus on how language works. Formalism studies how literary language communicates truths in stylistic or strange ("defamiliar") ways. Structuralism studies how language works in general and in a scientific way. The key figure here is Ferdinand de Saussure. He provided the logic of signification. A thing or idea is a "signified" and that thing or idea is represented by a visual or audible word called a "signifier." The signified/signifier relationship is called a "sign." Saussure believed that by understanding the rules governing signs, we could understand the laws that govern language in all cultural domains—not just literary domains. This is where structuralism differs from formalism.

Saussure saw structuralism as a much broader, audacious attempt at studying all aspects of culture by looking at language. But both formalism and structuralism share an intense interest in language itself and particularly "how" language works.

What is the relationship between formalism and structuralism, and how did formalism contribute to the development of structuralism?

Formalism and Structuralism both emerged in the early 1900s. Formalists focused on the texts of literature. They reduced the importance of cultural, biographical, or historical influence on a literary text. Instead, they focused on the form of the text: the grammar, the syntax, plot structure, roles, archetypes, characters. In other words, they focused on the structural characteristics that define a text as literature or poetry. They wanted to show how literature was different from other kinds of writing. So, there is a structural component to their literary analysis. 

Structuralism is a bit more broad and has been applied to other fields such as psychology, architecture, sociology, and anthropology. Structuralists believe that there is an abstract structure of language and culture. How that abstract structure is enacted in human life depends upon the interactions of humans themselves. For example, German culture and language are different from Chinese culture and language but things like grammar (subject/predicate) and certain interactions (economic, social) are similar. For structuralists, this shows that although cultures and languages are different, there are basic constant, abstract structures that apply to all languages and cultures. 

Formalism and Structuralism both emerged and influenced each other at a time when science (Einstein - early 1900s) and sociology were growing in importance. So, those of the arts, humanities, and human sciences wanted to make attempts at proving the scientific merit of their respective fields. This was both a strength and a weakness for Formalism and Structuralism because while it gave them some scientific credentials, they did so by ignoring the historical and personal aspects of culture and the arts. 

Formalism and Structuralism share this idea that for a certain area of human life, there are abstract rules that determine certain functions. Formalists have a set of criteria for literature, a set of rules and structures that determine what is literature. Structuralists have sets of rules and structures that determine how a language works and/or how a culture interacts. One could say that structuralism is a broader application of the Formalist idea that an event or an object (such as a literary text) is defined (and for Structuralist, determined) by some abstract set of rules. Both Structuralists and Formalists have been criticized for reducing the role of history (social and political) in their analyses of literature, language and culture. Both movements aimed at being scientific.