Hardware encoder for live streaming?

I was looking at the Blackmagic H.264 Pro Recorder or the Decklink Studio. I'm trying to figure out live web streaming for the first time.

What I'm unsure about is whether I need one of these devcies to first encode to H.264 and then bring it into the computer or whether Adobe Flash Media Live Encoder does the on-the-fly encoding itself. If the Adobe software does the encoding itself, what are these two devices used for, just getting the footage for the camera to the computer? Do I need a hardware encoder?