I'd definitely keep the special ones from and to close family.
I'd say don't be too hasty to bin/recycle the ones from parents or grandparents, as they can take on a lot more significance when the person is no longer with you.
I'll probably be called a saddo by most people, but I always write the year (small) in the bottom right corner of the left side of a card to somebody close. If it gets thrown away/doesn't survive, then so be it and nothing lost; but I like to think that, if it does get kept, knowing when it was given and received may just be of some sentimental interest to somebody years or decades from now.
I have all of the birthday and Christmas cards that my now long-gone parents gave me, but apart from early ones with age numbers on the front, the only one I know for sure when it was from is the one that's just signed from my Mum, rather than from both of them, as she lived one year longer than my Dad.
I'd love to be able to look back and match them to memories and photos from that time, to see what we were all doing back then at that snapshot of everyday but glorious family life.
Whether internationally groundbreaking or quotidienne and personal, all history has to be actively made - and preserved - in the first place before you can ever have the opportunity (if desired) to look back on it in the future.