When your link pointing on other website than its your back link . Google does not give value to all back links.
one reason can be dofollow link changing to nofollow. another reason can be the page where your link is on moved to archive and google can't find that page anymore. there could be many other reasons. some site owners practice link exchange in a dishonest way and while they get dofollow links from their partners, they set their link page with noindex/nofollow meta tag to keep them safe from google penalty. even though the link itself doesn;t have nofollow attribute, the entire page has nofollow/noindex robot tag so none of the links count. just a few idea but there are various reasons for the links not to show