They are very similar.
Generates a relative/absolute path :
path('contact')
will generate /contact
Generates a scheme-relative/absolute url, ie domain + path
url('contact')
will generate http://example.org/contact
The url() style is useful when using cross-domain ajax or generating emails, because the hostname won't be the same.
Take a look at the code https://github.com/symfony/symfony/blob/master/src/Symfony/Bridge/Twig/Extension/RoutingExtension.php for more information
url Twig function generates absolute path path Twig function generates / related url
Example we have http://sf2sandbox.local with AcmeDemoBundle
{{ path('_welcome') }} produce /
{{ url('_welcome') }} produce http://sf2sandbox.local/