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Hi Matt,
Thanks for asking.
The built-in IR Receiver (and thus the Apple Remote) will give you a
snappier experience, whereas the EyeTV Remote will give you many more
buttons.
With a universal remote, of course, it's very easy to combine the best
of both worlds, i.e. using the Apple Remote buttons as navigation
buttons and the EyeTV remote buttons for shortcuts to some dedicated
functions.
You can also use RB's support for multiple Apple Remotes to
differentiate up to 255 x 6 buttons. A user put together a nice
tutorial for Harmony users a few months back. You should be able to
find that tutorial in the forum by searching the forum for "Harmony".
Best regards,
Felix Schwarz