A hardware firewall will only see outgoing traffic from a trusted source (your computer) and let it all out without a blip.
That's not entirely true. A hardware firewall, properly configured, will see if your computer starts trying to send traffic out on an unauthorized port or to an unauthorized IP address. It's called egress filtering, and it should be used more than it is. Some firewalls, Application Proxy Firewalls in particular, can even tell if you are sending malformed packets on an authorized port/protocol.
That part that is true is that a hardware firewall won't be able to tell if it is Internet Explorer or some piece of spyware on your computer that's doing the communicating, if it's sending its info over a valid port/protocol, to an authorized destination, using valid packets.