FastCGI is faster than mod_* in general, unless you're a python shop, then wsgi is probably better.  It looks like the php-equivalent project is php-fpm.

FastCGI is highly optimized C-code specific to dishing out dynamic web requests.
mod_php is just the php interpreter embedded into apache.

Another nice thing about FastCGI is it lets you spawn/control multiple worker threads, so it tends to be better for scaling-out your app, and you can redirect traffic to workers that aren't necessarily on localhost and use apache as a faux load-balancing solution.