I've recently been asked a number of times what blogging software I use, and why I use it. To answer these more fully, I use blosxom with a custom design.
I have two problems with most blogging software. The first is that blogs are static and are usually presented as dynamic. Blosxom can run in two modes, static and dynamic. In dynamic mode it is a cgi application which operates much like any other piece of blogging software. In static mode it compiles the site and publishes it to a chosen directory. I use both modes. The majority of this blog is static html which is updated every 5 minutes (only the changed bits). If you go to the writeback bits, that is cgi (so you don't have to wait 5 minutes to see your comment, or other folks' comments).
The second problem I have with most blogging software is that it is remarkably difficult to publish, edit, and work with stories. Blosxom uses a plain text backing store arranged on the filesystem. You can grep it. Rublog has a potentially better system where it will talk directly to CVS, but I've been fine with svn'ing my local repository and rsync to publish. For things like Java, working with a relational database is usually easier than the filesystem, which says something =) Personally, I much prefer working with files for stuff like this. I want to make a change, it is just a vim away.
Some of the things I like would be considered huge drawbacks to other people. Like most compiled vs. interpreted things, static site compilation is a touch trickier -- I had to be able to set up httpd to cleanly switch between the two (not difficult, but trickier than not having to). The benefit is that the blog should be rock solid and fast (340 times faster than the CGI'd version).
Additionally, you have to know how to, and be able to, set up a cron job. Actually, you could ping the cgi script to do it, but cron is easier =)