I have worked with both. Highly recommend the Hauppauge.
The BM can record through component cable and HDMI, and is installed in your computer.
You get a component cable with it, but if you want HDMI, you'll have to get the cable yourself.
The recording program comes with a nifty screenshot feature.
I really liked this card, but unfortunately, it broke. The customer service is excellent, I sent the card to them, and I was given a replacement for free. However, that one didn't last longer than a bunch of months. It might be me being unlucky, though.
After the 2nd one broke down, I got the Hauppauge HD PVR (not the gaming edition, btw), and it's been working for ages now and like the BM, it's very easy to use.
The Hauppauge records only through component cable, and records via USB.
Everything you need is included in the box, so you don't need to purchase anything else.
Unlike the BM, the recording program does not have a screenshot feature, but most video editing software have these, so I can't say I find it a loss.
My setup enables me to watch the feed on my computer and on TV as well. In the case of the BM, I could switch off my TV and just watch the recording feed on my computer. The Hauppauge however, displays a delayed feed on your screen, by about a second, so if you want to record gameplay, be sure you can watch the feed 'live' on your TV.
Neither of these requires a high end computer.
Other than that there's barely any difference. I could let people see footage record by both, and you couldn't tell which I used for recording.