I've never used python before. I've used php for about 5 years now. I plan to learn python, but I'm not sure what for yet. If I can think of a project that might be better to do in python, I'll use that to learn it.
Edit: just to add this as an important note, I do mean strictly for linux, not multi-platform.
Edit 2: I'm hoping for objective answers, like a specific project, not a general field of projects, etc.
PHP for websites. Python for pretty much anything else, such as commandline tools, long-running scripts, daemons, etcetera. If you're writing a PHP script and you're reaching for functions in the posix extenstion, shared memory or other low-level stuff then that's generally a sign that Python would be better suited. It's not that PHP can't do it, but Python just does it better and less buggy.
Especially when you're venturing into background daemons for your website you'll want to look at Python. PHP has some garbage collection problems in long running processes such as daemons. Also, some functionality is much easier and clearer in Python (e.g. redirecting STDIN, STDOUT and STDERR. PHP misses posix_dup2()). Also, Python has threads :-)
The only time when I now use PHP background daemons for my websites is when they can re-use significant amounts of code (such as with MVC frameworks like CakePHP).
One more advantage of Python is that there are many, many libraries for it, because it's rather easy to create a Python wrapper for a C library. So, Python has libraries that PHP doesn't have (OpenGL, multimedia, etcetera). So if you're into those areas Python becomes the obvious choice.
From a coder's perspective, what kind of project should I choose python over php for where both could do the job?