I had a six digit ICQ, and I knew some people with 5 digits, so you could instantly tell how cool people were. Also, ICQ would say uh-oh when you got new messages, and AIM just dinged.
On a technical side, ICQ did offline messages forever and AIM didn't do offline until after AOL bought ICQ. ICQ also needed UDP to work and did peer to peer messaging when possible (coordinated through the central server) and I think you could send files sooner. Otoh, AIM servers responded on all tcp ports, so if you had a derpy corporate firewall that just looked at ports, and you had any ports open to the internet, you were golden; the java applet client was convenient too.
I think ICQ is still around, AOL sold it to some Russian company a decade ago, but AIM is dead; Verizon didn't want to keep it running, or Yahoo Messenger either.