Formalists, like the New Critics, focused on the form and structure of the literary text; its literariness and style. They did not look at extrinsic factors such as authorial biography, historical context or politics. Their goal was to find the formal structures and techniques that make literature different from all other forms of writing: tropes, symbols, archetypes, and the different stylistic traditions among different genres. For example, the form of prose vs. the form of poetry; they would start here and get more nuanced and complex. Formalists employed a method called ‘close reading’ which focused just on the text. The goal was not so much to discover what a text meant but to distinguish what was literary and what was not. Any analysis which focuses just on the text will employ a kind of close reading. Similarly, a linguistic analysis of a text will be a close reading but the linguist will go beyond the present text and look at histories of etymology and language to supplement their investigation; and this may also include historical/anthropological research and research of other related literary works. Linguistics is a study of language; not limited to literary analysis.
Structuralism is much bigger program and it is more scientific and interdisciplinary than linguistics or formalism. Again, you could do a close reading with a structuralist agenda but your goal is to uncover the underlying structures and system of signs (semiotics) which consciously or unconsciously influence rules of culture. Such roles are said to be cultural or social constructs. This means that the individual and anything he/she creates is always already influenced, sometimes unknowingly, about how to dress, what to eat and what literature is. There is some similarity here with formalism in the sense that both structuralists and formalists look for the common elements of what is used to build or make literature. Structuralists go beyond the literary text itself to look for the deep structure that forms all kinds of discourse, language and linguistics; not just literature.
Formalism, structuralism, close reading, and linguistics are all similar in their close focus on styles, common elements, structure and form of different kinds of writing as it applies to assessing what makes literature different from other kinds of writing. Where they primarily differ, is what they consider part of literary analysis. Formalists stick to the text and that’s it. Linguists go a bit further and look at historical etymology and language. Structuralists are interested in the structures that motivate all discourse, so their program is more involved and sociological. Close reading is a practice of scrutinizing a text and it is formal structures and it arose in the New Critic/Formalist schools. But the concept of close reading, in general, can refer to a detailed analysis of any text and in this less used, general sense, close reading can be supplemented by other theoretical approaches such as Marxist or Feminist.