Old walls move. They were built to move — soft stone, soft mortar, the whole assembly flexing with the seasons like a hull. Point them in cement and you put a rigid crust on a moving body: the wall keeps moving, the cement does not, and the stone spalls off its own face a winter at a time.
Lime is slower, softer and more expensive to lay. It also takes carbon back as it cures, forgives movement, and lets water leave the wall the way it came in. We stopped arguing about this in 2013. The walls we repointed that year have not said a word since.