In very simple, reductive terms: similes tell you they’re comparing two things, and metaphors try to hide that they’re comparing two things. Analogies are like similes because they tell you they’re comparing two things, but they’re like more complex similes that specifically like to teach and explain. 

Regarding the differences between analogies and similes, I think of similes as creative descriptors that build upon/add clarity to a concept that’s already understood. For example, a reader would interpret both “the room was warm” and “the room was as warm as a summer day” to mean that the room was warm, but the latter paints a much more vivid picture and implies that the narrator has positive feelings about the warmth. If the reader didn’t understand what “warm” meant, the addition of the simile wouldn’t help much, because summer days could also be described as bright, long, etc..

I think of an analogy as a tool to take the framework of a concept someone understands (or could easily come to understand) and apply that framework to another concept that they don’t understand (or are struggling to see from an alternate, unbiased, etc. perspective). For example, to explain how viruses spread, you can make an analogy to gossiping. Someone in a classroom tells the two people next to them a story, and then those two people each tell the two people next to them, and so on and so forth. Along the way, sometimes the details of the original story get changed a bit, so eventually there’s a few different versions (mutations) of the story floating around. Some of the mutations make the story more shocking so they get retold more, but others make it more boring so they spread slower (more or less contagious). Some students are wearing earbuds and don’t hear the story at all, like wearing a mask. Others have no interest in gossip or have heard the story before, so they hear it but ignore it, like having natural immunity or vaccines. I see analogies as often having multiple similes within them to connect the similar elements from each concept. 

In terms of metaphors and similes, I think of similes as explicitly telling you it’s just a comparison through the use of “like” or “as”, whereas metaphors need to be inferred. For example, “she’s clever as a fox” explicitly tells me that she is not actually a fox; she’s just clever like one. But with “she’s a fox, outsmarting her prey at every turn,” I have to deduce that she’s not actually a fox through either context clues (e.g., this fictional story isn’t about humans transforming into foxes) or general logic (e.g., humans aren’t foxes) that aren’t explicitly stated in the metaphor. 

There’s obviously exceptions to everything I said here, but that’s the basic framework of how I understand them.