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

The terms "figurative language" and "figure of speech" are related to a certain extent. However, they are not terms that should be considered as interchangeable. 

It's best to consider a "figure of speech" as a smaller part of what might be considered "figurative language."

Figurative language is essentially the antithesis of a form of writing known as "realism." Realism disposes of the tropes of metaphor and simile to (hopefully) let the reader know that what they are reading is literal. When a piece compares a person or thing or place to another person or thing or place, the writer is using a metaphor, common in figurative language. Look for the words "like" or "as" and you have likely encountered a simile which is a clear indicator of figurative language.

A "figure of speech" on the other hand, does not necessarily indicate figurative language. In many realist dialogues, "figures of speech" are used to drive plot/character development/tone. A completely realist piece of writing may contain many figures of speech, usually in the form of dialogue. Without context, a reader should deduce the time and place of a story or article through a figure of speech.

For instance, a piece that forgoes figurative language could use, say, the colloquialism, "square," in place of it's literal meaning, "cigarette," and the reader would understand that said piece is set in an earlier time period.

It's best to consider these terms as two separate entities but helpful to consider that figures of speech may not always play a role in a work utilizing figurative language. I've never heard nor read anyone saying that the object of their affection was as "dizzy as an old basketball." This is a simile indicative of figurative language as it is not a commonly known "figure of speech." I know this because I just made it up.

Now, to call someone or their actions "sweet as pie," is commonly known as a figure of speech. This doesn't mean that it's inclusion in a piece of writing indicates that the piece consists of only figurative language.

In short, figurative language utilizes a writer's creativity by use of simile or metaphor to (ideally) achieve an apt and original comparison. It is possible to create one's own lexicon, a figure of speech is usually well-known, recognized phrase.

Hope this helps. Best of luck.