In poetry, line breaks are powerful tools. They are natural stops to the eye and mind, so they become a form of punctuation, a way to separate ideas or images, all without needing a symbol.
Line breaks are useful in non-poetry, too. This is one of the lessons of design and layout a writer should learn. Even in prose, line breaks have power. If you are writing a document, do your best to break at sensible parts. Try not to break up a proper name. If it is an organization with several words in the name, at least try not to leave just one of the words on one line. (For that matter, don't leave widows or orphans in your writing, either.)
This is the final coat of polish in writing. Your words are down, they're not going to be changed anymore. It is all about fine tuning margins and manual page breaks there. I don't expect it to be done in a massive work like a major novel (though I would appreciate it). But if you are writing something 5 pages or less (if not 25), it is inexcusable.
Line breaks matter. They will matter in every form of writing you do. They affect the eyes, which affects the mind, so no matter how subtle it may be, it is a tool that cannot be avoided, and should not be ignored.