There is only one major concept for which being an Am Ha’aretz is relevant (well, two). The main one is that Amei Ha’aretz are considered T’mai’im for certain things. The second is for Terumos and Ma’asros.
Interestingly enough, I was listening to Rav Schachter’s shiur online last night right after reading this thread. (Mods don’t let links to YUTorah, but it’s titled Shiur #21-Yevamos-Chazara under R’ Schachter.) He quotes that some people still hold that we are all Amei Ha’aretz Bizman Hazeh (because no one accepted to become a Chaver in front of 3 Chaverim) and that he was once by a Rosh Yeshivah in E”Y who wouldn’t give him anything that needed Terumos and Ma’asros taken off because no one can trust anyone else Bizman Hazeh for Terumos and Ma’asros.