They will both handle those things, but the 37's will be a little bit more comfortable and of course they provide more traction (given the same tire) because they have more surface area in contact with the road.

The advantage to the 700x32 tires is that they will be a little bit faster, although it's more of a subtle difference. 28 vs 37 is a bigger difference, but 32 vs 37 is pretty subtle. You'd probably have to ride them back to back to notice the difference.

Edit: By off-road I didn't necessarily mean 'mountain bike' type terrain, I meant dirt or gravel trails.