Brendon cited some advantages in another thread. I think it boils down to this:
Anthem.NET
Pro: Easy to install (xcopy), easy to use (e.g. change <asp:Panel> to <anthem:Panel>)
Con: Few fancy controls like pop-up dialog box, little 3rd party support (e.g. ComponentArt has not created an Anthem.NET-compatible version of their controls)
Microsoft ASP.NET AJAX
Pro: Several fancy controls like pop-up dialog boxes, 3rd party support (e.g. ComponentArt has created a Microsoft ASP.NET AJAX-compatible version of their controls)
Con: Hard(er) to install, hard(er) to use
Anthem.NET is good enough for most web sites. And you can use both Anthem.NET and Microsoft ASP.NET AJAX on one web site. If you have control of the web server (like Brendon does) you can add Microsoft ASP.NET AJAX and use it when you customize the site.
I have been using Anthem.NET and
Yahoo! User Interface library together to build my
shipping provider editors. YUI gives me modal dialogs, drag and drop, pop-up calendars, etc. without any installation (the entire library is javascript
that can be hosted by Yahoo!). Anthem.NET gives me all the nice Ajax callback goodness. It seems like a pretty good combination.