Sorry for my late reply, just found this thread. Anyway, realistically the only difference would be the potential continuation of the Stuart line into the 19th century.
@jb3, Prince James wouldn't be marrying a Saxon Princess, that's for sure. WAY too high ranking. Plus by the time James is marrying the Jacobite movement would have collapsed and James III would be aiming to secure his sons futures in the Papal states. So likely he looks for a wealthy Italian heiress from a family that see the Stuarts pretendership as more important than a match with money or lands. Now one can use the birth of a third son to start butterflies that would result in a Stuart Restoration, like your suggesting, but I don't see that happening. The 1759 invasion attempt was after the Stuart chances had died and a mere year before the accession of George III, a wholly English Hanoverian. And the Highlanders were entirely done after 1746, so no chance of a '59 rising. Really the Jacobites last chance was the '45 rising.
The only potential chance of a Stuart restoration coming out of this would be if Prince James's grandson married the TTL equivalent of Charlotte of Wales or Queen Victoria.