His name appears in dozens of song lyrics[1], many of them from after his death. Many are in other languages, and he's often paired with a similarly famous female star (Bette Davis, Audrey Hepburn).
I suspect many people have heard of him even if they can't name a single film he was in.
Ha! First, let’s not forget. Grant was a “movie star”. But seriously, he was a star from a time even before I was born.
The difference probably has to do with the difference between people who grew up during the ages of broadcast television, cable television, video rentals (prolly add PC video games here), internet entertainment, and internet ‘television’.
If you grew up in the age of broadcast television you watched everything or went outside.
As an aside, I was talking to an intern at work. Somehow movies were mentioned and I guess I cited a quote from Pulp Fiction (1994).
Me: Oh? You haven’t seen it? It’s a great movie.
Intern: I don’t watch old movies.
(>_<)
I actually know Cary Grant very well, you could say I'm a bit of a fan. When DVDs were a thing (giving you a hint about my age) I bought lots of his movies. So I can truly second the recommendations mentioned in the other answers here.
But I wouldn't have been surprised if knowledge of someone like Cary Grant couldn't be generally expected from the average tech audience of today, although I believe that they are on average more knowledgeable about and interested in general (everyday) culture. I mean, there has been more than one instance where I heard a younger colleague talk passionately about how much they liked a certain movie or show which I also love - only to find out that they are actually talking about a recent remake that I had never heard of. So not everything can be taken for granted.