What is the difference between an analogy and an idiom?

Short answer:

To explain something with something similar is an analogy.

To mean something totally different than what you are actually saying is an idiom.

Longer and more explaining answer: To explain one thing by using something different yet similar as a maybe easier "example" is an analogy.

We can crudely explain electricity and a light-switch like this:

"Electricity is like water. When you turn on the water, it starts flowing.

When you turn on a light-switch, electricity starts flowing"

An idiom however is very different. It is a common saying that does not mean what it actually say.

Examples of this is:

1 A piece of cake.... ---> We say this to indicate that something is easy.

2 To wake up with the wrong foot.... ---> Person being grumphy.

In my opinion, an Idiom is also a kind of analogy but only because we are taught the added meaning of the Wording.

An example here is that foreigners of a country do not easily understand the local idiomatic expressions. This because idiomatic expressions are mostly local to that country alone. A huge exception is all the English speaking countries who share a lot of idiomatic expressions.

Foreigners need to be taught the expressions in order to understand their meaning.

An analogy is something that normally would works universally with no barrier of language or country.

There are however other barriers when using analogies.

The key here is knowledge.

A good analogy will explain a difficult subject in easy terms using only phenomenons or science the listener can relate to.

Analogies can be made up on the go...

: She was as clever as a duck drowning.

---> Not very clever of a duck to drown.. Ergo, She was not the brightest one.

: The paint on that house is as good as the paint on my scrapped car.

---> We can count on that the car didn't look very good. Ergo, The paint on the house was not much good either.

Idioms can not be "made up on the go" normally.

It is only when many people recognise a sentence or a particular Wording as having a different meaning that it becomes an idiomatic expression.

There are some idiomatic expressions that are also an analogy. Not that many though.