So I've been using Horde as my webmail interface for 10 days now, and I must say, I really don't have many complaints. It may not be as elegant or snazzy as Gmail, but it works just fine, and I know that the Googlebots are not reading my mail and instantly deciding what to sell me next.
The mail interface on Horde is just fine. They do have an Ajax version, but they don't have all the modules ajaxed yet, so the Ajax landing page can't show your task list and notes, at least not in the version my provider has installed. It's not displaying HTML email inline, but that appears to be a configuration issue with my provider, and not a bug in Horde. It works with Spam Assassin to filter spam, and it works great. I have Spam Assassin set up to auto-delete anything with a score of 7, and to filter to the spam folder anything between 2 and 7. I'm getting 1 or 2 spams a day that make it to the inbox. The mail app is full of nifty touches, like the "reply to list" option for mail from a listserv. That is something that even Gmail doesn't have.
One really cool thing about Horde is that the portal page can show your unread mail count in each mailbox, your upcoming events from the calendar, your task list, and notes. You can also add weather, news, RSS feeds, etc. It's pretty slick.
The calendar module is perfectly functional. Again, not as slick as Google or Yahoo, but good enough. It does reminders, and you can subscribe to remote calendars. One thing that is lacking is that there is no mobile access to the calendar yet. They do have a sync module, but my web host does not have it enabled, so I have not been able to test it.
The notes and to-do list modules are simple, but they do what they are supposed to do. And having everything visible on one web page is really very useful. One nice touch is that you can move or copy an email to the task list.
Horde is very, very configurable. In fact, I'd guess the options menu is probably overwhelming to most people. It pretty much is the default mail client that comes with just about all web hosting and email accounts. It's also used by many universities. It is open source, the developers appear to support themselves by doing consulting related to large scale Horde implementations.
I don't see any reason that I can't be content using Horde as my primary mail and calendar interface.