I figure when they were drawing up the plans for the Mountaineer, they asked "how can we take a gun like the LRH and improve on it?" So, all things being equal (espically price), I would choose the Mountaineer. That said, I own a LRH that I converted to bare primer and have a Lehigh converted breech plug and shoots BH 209 well. Like the set up.
Accuracy is accuracy. Probably can't tell one from the other in that dept.
As some others have stated in the past, the stock on my LRH is more designed for the bench than packing up the mountain, so if that is what they still sell for a stock on the LRH, the nod would go to the mountaineer in that dept.