XAJAX uses XML as a transport for data between the webpage and server, and you don't write your own javascript data handlers to manipulate the data received from the server. Instead you use a php class and built in javascript methods, a combination that works very similiar to the HTML_AJAX_Action class and haSerializer combo. XAJAX is designed for simplicity and ease of use.
HTML_AJAX allows for multiple transmission types for your ajax data - such as urlencoding, json, phpserialized, plain text, with others planned, and has a system you can use to write your own serializers to meet your specific needs. HTML_AJAX has a class to help generate javascript (HTML_AJAX_Helper) similiar to ruby on rail's javascript helper (although it isn't complete), and an action system similiar to XAJAX's "action pump" that allows you to avoid writing javascript data handlers if you desire.
But it also has the ability to write your own data handling routines, automatically register classes and methods using a server "proxy" script, do different types of callbacks including grabbing remote urls, choose between sync and async requests, has iframe xmlhttprequest emulation fallback capabilities for users with old browsers or disabled activeX, and is in active development with more features planned (see the Road Map for details)
HTML_AJAX has additional features such as client pooling and priority queues for more advanced users, and even a javascript utility class. Although you can use HTML_AJAX the same way you use XAJAX, the additional features make it more robust, extensible and flexible. And it is a pear package, you can use the pear installer to both install and keep it up to date.
If you're asking which is "better" - as with most php scripts it's a matter of taste and need. Do you need a quick, simple ajax solution? Or do you want something that's flexible, extensible, and looking to incorporate even more great features? It depends on the project, you as a writer, and your future plans.

As of 0.3.0, all the examples that ship with HTML_AJAX have been verified to work with
* Firefox 1.0+
* Internet Explorer 5.5+ (5.0 should work but it hasn't been tested)
Most things work with
* Safari 2+
* Opera 8.5+

Once all the major features are complete and the API has been tested, the roadmap gives an idea of whats left to be done.

If you run into an HTML_AJAX problem only on some servers, chances are your running into a problem with output compression. If the output compression is handled in the PHP config we detect that and do the right thing, but if its done from an apache extension we have no way of knowing its going to compress the body. Some times setting HTML_AJAX::sendContentLength to false fixes the problem, but in other cases you'll need to disabled the extension for the AJAX pages.