Why `textwidth` gives me the same effect as `linebreak`?

Question

Could anybody explain, why textwidth works gives me the same effect as linebreak?

Example 1. If I type aaaaaaaaaaaaaaaaaaa several times, the last aaa... is wrapped to the next line. linebreak is off, so why does it happen?

set number
set wrap
set textwidth=79
set colorcolumn=+1

Example 2. Everything works as I expect it. When my reach the right side of screen, only a part of the last aaaaaaaaaaaaaa is wrapped.

set number
set wrap
set colorcolumn=+1

Answer 1

:h 'linebreak' has the following:

Unlike 'wrapmargin' and 'textwidth', this does not insert <EOL>s in the file, it only affects the way the file is displayed, not its contents.

But with :h 'textwidth', a longer line will break with <EOL> after whitespace to get that width.