Which is going to be more accurate on distance? The GPS or the Adventure?
As with many things, "It depends", Matt.
If the GPS loses contact (i.e. in a "city canyon" or under heavy foliage, when it relinks to the satellites, it will attempt to draw the distance from the last waypoint as the crow flies (straight line) to make up the gap. In that case, the CatEye would be more accurate.
Also (depending on the profiles available in your GPS), if you have the GPS set to "snap path to road", there will be some minor inaccuracies compared to actual distance traveled as shown on the CatEye. Error aggregation probably won't get you, but it it something I have to acknowledge when traversing featureless deserts.
Also, if the GPS is sometimes turned off during your ride, the CatEye will also be more accurate!
Otherwise, it is roughly a wash, in my experience, with the slightest of margins generally going to the bike computer 'cos it is measuring wheel turns directly, rather than extrapolating from "ping points" on the satellite.
Best,
Dan.