What is an "email client" in the first place?
The Wikipedia version of History of email indeed glances over the popular aspects.
In Bulletin Board System we had email in almost closed circles, sometimes wider networks and gateways into the real internet mail-system. (Example FidoNet). That started at a fairly popular level during the 1980s. On Amiga, Atari, Mac and PC this was as text-based as it gets, often menu driven. But also the smaller niches were catered for like C64, other "home-computers". But also on the "bigger iron" there were things like mailx (not even menu driven) or
elm (released in 1986, and even today perfectly usable 'UI').
But popular means graphical so that the dumbest user can point, click and make a mess of the place called internet.
Enter the Pegasus and Eudora in 1988/9. These programs were real clients with a GUI that popularised email to a really wider audience. Easy to use, nice to look at. If you were "connected".
That makes the difference here since AOL was really popularising "connecting to the net". The popular phrase for this "democratisation" of the internt was Eternal September of 1993. Like other middle men of the time (compuserve, eworld, etc) it offered internal mail-like messaging and a gateway to the real internet. But despite including a "client", it still was a middle man, and as middle men do, it was charging you as well. Making it cumbersome and any user depended on AOL to "get there".
Hotmail was not a "client" but a free service on the "real internet"; offered free of nominal charge ("free" in the sense as "if some offers you something for free on the net, you are the product being sold"). As real and direct connections to the net proliferated, direct access to a POP-mailbox got more convenient than going through the closed ecosystems of these walled garden providers like CompuServe or AOL. Some of these services were client based, but with a service on the WWW-web. Even that got increasingly superfluous in the eyes of casual users, requiring only a measly browser to get stuff mailed.
The advertising frenzy with AOL-discs produced to make landfills overflow in a scale several times bigger than VCS E.T. cartridges might have helped a bit. But the underlying infrastructure of the net available to home-users and adopted in businesses made the change much more attractive. Dial-up only ended when DSL came into numbers, so the change-over took from 1993–2000 in most developed areas.
For a while, many people had the impression that in order to "go online" one needed AOL, a reason deduced from watching TV or opening one's physical mailbox in front of a house (Youtube: AOL Commercial Video). But hotmail capitalised on being a viral campaign on the net itself:
Initially, each Hotmail message arrived in someone’s mailbox mentioning the words in the signature of each email: “Get your free email at Hotmail” at the bottom. By clicking on the word Hotmail, the recipient was taken to Hotmail’s home page where the free e-mail service was explained. Back in the nineties most computer users paid for Internet service which included one e-mail address. A big benefit of having a Hotmail e-mail address is the fact that subscribers can access their account from anywhere in the world, instead on their home computer. And the fact that there are no extra costs for each extra email address, Hotmail’s approach resulted in thousands of sign-ups each day.
So for Hotmail, this viral approach was extremely cost-effective and successful. While one of Hotmail’s competitor spent $20 million to attract less than half of Hotmail’s subscribers, it spent only $500,000 over a 2 year period to gain 12 million subscribers.
Really going from AOL-Client (Launched March 21, 1993) to the WebMail systems of old (first 1994: PTG Mail) and then H o t M ai l in 1996 to the GMail-Hype in 2004 we're looking at very different things. Just looking at the timeline:
A BRIEF HISTORY OF EMAIL APPS
and the volume of mails sent, expressed in changes of volume:
The History of Email
Of course, the total number of mails just keeps on growing, as ever:
The Shocking Truth about How Many Emails Are Sent