I played with both at Friedrichshafen. Both are good, but there's no clear leader. Some touchy-feely thoughts, rather than hard facts:
The Garmin user interface is very slick (especially after the button-twisting nightmare that was the GNS530) and the hardware has a good quality feel to it. The Avidine looks feels a little more plasticy.
The guy working the Avidyne stand got a little lost showing me a couple of functions. It may be because he was unfamiliar with a new product, or it may be the Avidyne is less intuitive / more complex design. He seemed quite knowledgeable otherwise, so my money would be on the latter.
One significant difference is that Garmin is touch-only, whereas Avidyne has a touch screen but can be operated from the buttons.
If you happen to be taking a GNS530 out of the old steed, the Avidyne is a drop-in replacement.
Something the Avidyne guy mentioned - the Garmin has no off switch, so you need it on a circuit breaker in case you need to shed load after an alternator failure.
I'm not overly taken with the Garmin. It's good, but it's not brilliant. There are a couple of features I'd change in an ideal world. But having seen both I'm leaning towards the GTN750.
Incidentally, is the new reg just a new N-number, or have you moved it from a European register to the US?