The significance of Schaeffer weeping is that it depicts both the character of Dr. Schaeffer and what moved him, but also the deep hurt that he felt over divisions in the church during the early splits over modernism (Religious Liberalism). If you are not up on the history, it started with the split from Princeton seminary by J. Gresham Machen in the 1920s and continued with various additional church and missions related splits from the mainline Presbyterian denomination. The splits did not stop there, later resulting in numerous other splits within the camps over Christian holiness, and other things. All of which deeply moved Schaeffer to the point of spiritual duress. For more on all of the splits I recommend reading Frame's paper called Machen's Warrior Children.
Now one can argue that a lot of these splits were necessary, especially the split from the mainline church. Yet it is also easy to see that considerable damage had been done to Christians by other Christians in some of these cases, in the cause of being right, which is itself wrong, and this is cause for lament. Right motives with wrong actions will always and forever be wrong.
The results of these deep feelings would eventually produce a crisis in Schaeffer, and out of that crisis came the work True Spirituality, which is at the foundation of all of Schaeffer's works and L'Abri. He further elaborated on this topic in a more succinct way in his work The Mark Of A Christian. So reflecting on Schaeffer's tears during this event should be at the heart of our studies. Understanding this event helps us to understand Schaeffer in the deepest of ways. Until we as a church understand the difference between saying "Truth & Love" and actually shedding tears over it, the problems will not be resolved. This was Schaeffer's heart and what disappointed him as time went on. The church has still not understood this balance, and until it does there will still be a need for teaching Schaeffer's works.