One reason IBM could make the transition from electromechanical (not analog) systems to programmable computers is that it wasn't really a disruptive change to their business model or to their customers. They were still renting out punched-card business machines. On the other hand, the switch to digital cameras totally messed up Kodak's film-based business model.

Another factor is that both Thomas Watson Sr. and Jr. were very strong, effective leaders for IBM (based on what I've read and heard).