As I understand it, php-fpm makes PHP pages respond faster by managing a farm of pre-loaded PHP interpreter processes so the end user doesn't experience the overhead of initialising the PHP subsystem, but once a PHP page starts executing it takes just as long as it would under "regular" CGI. It's a bit like running a Perl script under mod_perl - the interpreter doesn't run any faster because its bound into the web server executable, it just start up faster.
Am I right, or is there something subtle going on under the hood that makes it run faster overall? The reason for asking is that I have a PHP application that I want to run periodically (I run a 'wget' for its URL every X seconds). I'm not bothered about the response time because its periodic, but I am bothered about the execution time.
No, you PHP code itself will not run faster, it's still the same interpreter. You might have a small latency profit compared to normal CGI, but who runs normal CGI anyway? ;-)
Another small help could be the code caching, but that's only really useful if the script is run often.