These two fallacies are easy to mix up, especially for students in their first course of logic. Both involve language leading to multiple meanings. An amphiboly is a misunderstanding that results from unclear grammar. An equivocation is an argument whose definitions shift throughout the argument.
Often, these fallacies are used in humor. As I tell my students, the one place a logician welcomes a fallacy is in innocent humor (meaning, jokes that aren't used in a logical debate to distract or obscure meaning).
This is a famous amphiboly:
"One morning I shot an elephant in my pajamas. How he got into my pajamas I'll never know." - Groucho Marx
The first sentence is taken contextually to mean that he shot the elephant while he himself was in his pajamas. But it could be taken as saying the elephant was the one wearing the pajamas. Because of this grammatical possibility, Groucho is able to surprise us with the second line.
Here is an equivocation:
"I don't expect any trouble swimming here. The sign warns of 'man-eating sharks,' but I'm a woman."
The issue is not one of grammar, but definition. "Man-eating" means "human-eating," but to take the word as its meaning of "male" plays on the ambiguity.
The Beatles were masters of wit, and in their early press conferences would respond with quick rejoinders. Each of these is either an amphiboly or an equivocation:
Press: Does it bother you that you can't hear what you sing during concerts?
John: No, we don't mind. We've got the records at home.
Press: How did you find America?
John: Turn left at Greenland.
Press: Do you like topless bathing suits?
Ringo: We've been wearing them for years.
Press: The French have not made up their minds about the Beatles. What do you think of them?
John: Oh, we like the Beatles. They're gear.
Press: Some people have been calling your work "Un-American". How do you respond to this?
John: Well, that's very observant of them.
So, which of these depend on unclear grammar, and which depend on shifting definitions?
This can be a fun exercise for students that illustrates the importance of clear speech.
However, some may view the dissection of grammar and definitions as minutiae, and introductory logic classes should focus on syllogistic form or fallacies.
Yet, I would argue that it is inherently urgent and practical to understand the role of grammar and definitions in logic.
Here, here, and here. one can observe the issues of vagueness and ambiguity being discussed in regards to US legal documents. The first is an editorial musing by John E. McIntyre, the second is an important document by James Madison, and the last is a charged online forum.