Is "the way" synonym of "how"?

In constructions like the following ones, could the way be replaced with how? Is there any difference between them?

I like the way she eats peaches. The way he looked at her...

Can anybody provide examples of any phrase where replacing the way with how (or vice versa) changes the meaning of the sentence?

To me, the way sounds better, but when I use soulless Google to translate it to my native language, it tells me there is no difference.

Yes, it is synonymous. Both are stating the method of doing something.

The way John eats apples. (The method John uses to eat apples.)
I like how John eats the apples. (I like the method John uses to eat apples.)

Good question. They are practically synonymous in the forms you quote, and refer to the method or process by which something is undertaken. So yes, you could replace "the way" with "how" in your quotes and they'd mean exactly the same thing.

There are some usage differences: "how" is a question word, so it alone can be used in the interrogative: "How does she eat peaches?". "The way" is a noun, and so the sentence "I like the way she eats peaches" refers to the method itself as its object; you could then say "it's a good way" and your listener will understand. You cannot substitute "how" in that second sentence; "It's a good how" is nonsensical.