There was a facebook thread about this not long ago. I was actually surprised at the range of opinions that came up in the thread.
My personal opinion is that
the awards should convey with the car. My first Model A had trophies like that and, while they didn't do anything for anyone, they were nice to display with the car (at least in my opinion).
Quite a few people have the opposite and just as valid theory: it is the restorer who put in the blood, sweat, tears, and money to get the car to where it was and win the awards, so shouldn't they be the ones to keep them?
One person mentioned that they couldn't sell their car until they removed the awards from the radiator. I kind of found that tough to swallow...but there are people who really aren't concerned about an award-winning car, even if that car had previously won titles on a national scale.
Awards, for me, are kind of a pedigree with the car. If I had the option of buying two cars that were the same, but one was a Henry Ford award winner...I'd definitely opt for the latter. I like the fact that the car had been looked at thoroughly by a number of unbiased eyes and they basically gave their "stamp of approval" that the restoration was as close to perfect as you could get. Then again...I'm one of those people who believe if you're doing a restoration, you get it as correct as possible, even if the car's going to be a driver. But...that's me and my weird take on life, which is not the same as everyone else's.