In terms of functionality, it's basically for PHP what Rails is for Ruby, but unlike many other PHP frameworks, it's been designed with the platform in mind rather than trying to force Rails' concepts into PHP, as a result if which it feels "just better". This has given it an edge over competitors, which is why Symfony is remarkably popular among the more professional PHP businesses.
I think it's basically the only mature and popular surviving framework that dates from the blast of PHP frameworks that came to be when Rails got popular and PHP coders wanted that too.