It's likely a low to non-existent difference late in games. The place it seems likelier to impact the game is the first couple times through the order, because on average, NL pitchers get 2.1 PA per game. This compares to 3.9 PA per game for the 8th spot in the order or 3.77 PA per game for the 9th spot.
This suggests that over a season, starting pitchers would come up slightly below 12 more times per year batting 8th than batting 9th.
Not a huge impact, but it seems like there might be some significance.