One quirk of English is that the singular and plural forms of many words can refer to different things. For example
I like chicken (the meat)
I like chickens (the animals)
In your first example, vegan and vegetarian (singular) are adjectives used to describe a particular person or diet.
She maintains a strict vegetarian diet
The singular form can also refer to a person who follows the diet, but only if indicated by the context. Also in this case you usually need some determiner (a, the, this, that, etc.) :
Susan is a vegetarian, so please don't invite her over for "steak night".
I know this vegan who is also a body builder. I don't know how he gets enough protein.
Vegans and vegetarians (plural) always refer to people who follow that diet.
Many vegans like to do some kind of mindful exercise like yoga.
Although I've never been one, in my life I've known many vegetarians.
It's important to understand what you are talking about when constructing these sentences. Some options when talking about these as adjectives:
Vegan is not the same thing as vegetarian.
Vegan and vegetarian are not the same thing.
Vegan and vegetarian are different things.
If talking about these as people, you have to write the sentence differently
A vegan is not the same thing as a vegetarian.
Vegans and vegetarians each have different dietary restrictions.
As for your other example, I would probably use quotes (or italicize the words) to make it clear:
"Chili" and "pepper" are different things.
Chili and pepper are not the same thing.
Although, actually this is not quite correct. It would be better to say:
A "chili" is a kind of "pepper", but not all peppers are chilis.