From a single PC, there's not really a performance advantage you will gain over game capture, but that comes with some caveats.
Game capture is great. It arguable works faster than directshow devices (of which a capture card like the Live Gamer HD is one) because it captures frames directly from GPU memory without using the CPU, whereas directshow devices have to use the CPU to get frames onto the scene. However, Game Capture currently only works with DX9 or higher games. Also, some games have hack shields that prevent you from hooking the game. Also, it requires a somewhat modern graphics card to perform well in order to copy the frames from the GPU efficiently.
Capture cards ignore all that, since they don't hook into the game at all. It is a simple capture of everything that is sent to the monitor. So a capture card is a lot like Monitor Capture, except it works with fullscreen games and doesn't require Aero to be off. It will show your whole desktop, not just the game (which could be a good thing or bad thing, depending on what you are trying to do). They are also resolution-limited: game capture can capture resolutions that consumer-level capture cards can't capture (such as 1920x1200).
So there are tradeoffs to both. If you want to have a dual-pc setup, you will definitely need a capture card, but if you are only getting it because you are planning to do a dual-PC stream in a couple years, I would definitely wait if I were you.
