The reason to have a email client is to send and receive all my emails from different. besides, the storage of email is very limited in the old days. usually using an client software can easy this problem but now this looks not a big issue. I will use Gmail as an example to demonstrate a web email is no worse than a Desktop client.
1. How fast you can see your emails.
I have using MS outlook and Thunderbird. they both take some time to launch and need more time to check new emails. they will be fast once start up but if you are the person don't like to keep them up all the time (like me ), you will sometimes get crazy if you only want to check an existing email because the client takes 10 seconds to initialize. with the web mail, it only takes 1 second to open the webpage and you will be able to see your inbox. I almost cannot feel any delay when I'm using Gmail. the interface is very simply and basic. the search function is also very quick sometime even faster than my desktop client.
2. See emails in one place.
This is the main advantage for an Desktop client. but today such Gmail and Hotmail both support fetch/send email from other account (if support pop3 ). for example, you can see the Hotmail emails in your Gmail. and you can also send email from Gmail with its sent address to be set to your Hotmail address.
3. sync your contact and calendar
Actually web clients are more friendly as they are synced by nature. Desktop client won't be able to do this unless connect to a server (like exchange ).
4. MS Exchange !!
Here we come to the main challenge. for some reason, MS exchange and Outlook look like in a dominating position. this combination offers email, calendar and direction Service (contacts) together. this was too powerful than the web mail in the past. but nowadays, calendar and contacts have become very common in a web mail. the main problem is no big company can suffer such a transfer because exchange is totally a different format.
Recently I have seen universities moving their student email system to big company services (I. e. hotmail, Gmail ). this is good for both universities and students. as universities don't need to maintain the system and students can enjoy the modern interface (compare to the previous email system ). but staff email system remain the same (exchange ).
Maybe in the future, all mail will move to web based?