What is the difference between figurative language and a figure of speech?

Figurative language is a broad category that includes figures of speech as well as sound devices and imagery. Figures of speech are words or groups of words that must be taken in a non-literal sense in order to understand their meaning. Examples of figures of speech are similes, metaphors, irony, synechdoche, allusions, puns, hyperbole, understatement, and personification. All of these use words in a non-literal, or figurative, way. 

Figurative language can also include wording that...

Figurative language is a broad category that includes figures of speech as well as sound devices and imagery. Figures of speech are words or groups of words that must be taken in a non-literal sense in order to understand their meaning. Examples of figures of speech are similes, metaphors, irony, synechdoche, allusions, puns, hyperbole, understatement, and personification. All of these use words in a non-literal, or figurative, way. 

Figurative language can also include wording that creates an image for the reader or that uses sound devices to add a deeper level of meaning to a word or group of words. Imagery is wording that evokes a five-senses response. Words such as "glittering," "screeching," "frigid," "lemony," and "smoky" create sensory reactions in the reader. Onomatopoeia, words that mimic the sounds they name, are a type of figurative language. Words such as "drip," "howl," "whiz," "tinkle," and "zip" are examples of onomatopoeia. Other sound devices such as alliteration, consonance, and assonance are also types of figurative language. Poetry especially takes advantage of such sound devices to create a mood or reinforce a meaning. For example, in these lines from "Stopping by Woods on a Snowy Evening," the repeated long /e/ sound reinforces the sweeping wind: "The only other sound’s the sweep
Of easy wind and downy flake."

So figures of speech fall into the broader category of figurative language, which includes words that are non-literal in their meanings as well as words that evoke sensory reactions in readers (imagery) or use sound devices to enhance the meaning of a phrase or sentence.