In theory, it is a sizable advantage. In reality, I'm not so sure it is. Last year the wild card winners were Baltimore and St. Louis. Baltimore lost to the Yankees in 5 games, the last 4 of which were ridiculously close. St. Louis knocked off Washington and had a 3-1 lead on the Giants before blowing it.
The wild card winners may be a little more tired than the teams they play in the LDS, but they may be sharper too because they've played more meaningful games recently. I think most of it washes out.