(1) Less advertising.
(2) Faster (easy to do if there's less advertising and graphics).
(3) Make it easy for users to create throw-away addresses on the same account that they can use for ordering things online if they're worried about getting spammed. For example, don't allow hyphens in account usernames, and then allow any user to create virtual addresses by adding a hyphen and then something else. e.g, if their account is "email@example.com", then they could use "firstname.lastname@example.org", "email@example.com".
(4) Build in features not commonly associated with an email provider. For example, a calendar/reminder service that automatically emails the user about events they enter.
Maybe some of these things are already in modern webmail, it's been a while since I've used webmail. I'm pretty sure #3 hasn't been implemented yet, though. That could be your Killer App. When you get rich, make sure I get a percentage. :)