Not really any difference mechanically. Even the problem rifles for the most part had the problem corrected so I wouldn't be too concerned about it. I personally prefer walnut stocks because they are all 1943 and earlier. Not that they were used in the war but they were carried in defense during the war. Plus, they generally look better than the birch because the birch did not hold up as well to the typical stock abuse. Walnut stocks also generally are worth a bit more. Not enough to get excited about but a little.
What shoots better depends more on the wear of the bore. I've never seen or even heard of a shot out Swissrifle but that doesn't mean they don't exist. They were shot regularly but I don't think in massive amounts to cause excessive wear.
I might be concerned with grease in the bore because it could be hiding a problem. I would ask to remove the grease prior to purchase. If he says no, I'd like to know the reason and have a return policy in writing before purchase.
I've never seen one with a grease packed bore either. These were used, they were kept in the home and they were shot for qualification purposes on a regular basis. If it is a worn, well used looking rifle packed with grease, I would question it. If it is a bright, shiny, new rifle, not so much. May have never been issued.