I like both of them, they have their strengths and weaknesses. Lighttpd seems to be the better choice for serving downloads and files, Apache seems better for more robust systems. I did a little Apache vs Lighttpd on my own home server. I posted the results on my blog. http://blog.noobbox.com/2008/09/apache-vs-lighttpd.html Lighttpd had an overall quicker response time and uses less system resources, but some things aren't compatible with it like .htaccess files and scripts that depend on .htaccess for url rewrites (drupal, wordpress) although you can get it to work.
As it stands right now, Lighttpd and even more so nginx are the servers of choice when it comes to serving static files. Both are extremely fast and can serve images and other static files much much faster than apache and with much less memory use. Apache is still the king when it comes to PHP sites and many other dynamic things.
True, lighty needs more development to compete in the dynamic serving area...it leakes memory a lot in that case. For static file serving, it's hard to beat.
My old boss did some benchmarking with Lighttpd and PHP and found it to be a big improvement. Results for Lighttpd with PHP and Lighttpd with Symfony There are some downsides but they seem to be significantly outweighed by the upsides. That said, we never got around to actually implementing the switchover to Lighty on any production servers.