FYI is when the recipient has to know this piece of information, usually they didnt know any of it, or at least some of it, already. And it was important that they know this, because it will be necessary for them to perform their function, e.g., informing support team about a feature change to answer customer issues, or telling your production team about a code change that may potentialy cause performance issues.
FYR is a piece of content that is not so critical for the receipient to know, but it might come in handy. Usually, the recipient already knows the gist of it from prior conversations, but there is a chance that they may want to dig in more or refer to this piece of content. e.g., telling your boss in 1:1 latest status of an important sales lead, and later, forwarding to him the email exchanges you had with the contact at that company; or giving a presentation to your colleagues on a new infrastructure you developed, and later forwarding them the slides you used.