When Should I Redirect a Page After Changing Its URL?

I need to change a page URL to make it clearer, but the existing address has been live for a while and may already have backlinks. I’m unsure whether a redirect is always the right move or whether there are cases where keeping the original URL is safer.

How do search engines and visitors handle the old URL after a change? I’d like to understand what to consider before publishing the new address, especially when changing a page URL that already receives traffic.