A short answer to the question "What is scientific Christianity" might be "It's the Christianity Jesus practiced." In other words, it's the appearing again in human experience of Jesus' kind of Christianity. It isn't something different from Jesus' Christianity or something added. The clearer we are on this, the better it is for us and for our practice of Christian Science.
No one could improve on Christ Jesus' life and teachings; Christian Science certainly doesn't try. A mistaken sense of improving on the Master's teachings—for example, supposing that we were making them more systematic or exact—is what would ultimately make one think that Christian Science is a cold abstraction or unworkable theory or in some way unlike the spirit of the Comforter.
The fact is we don't do something to Jesus' teachings; they do something to us. We are changed by living with them—as Paul says, "changed into the same image from glory to glory, even as by the Spirit of the Lord."II Cor. 3:18 There is really no way to do Christian Science without becoming a Christian Scientist, and that inevitably means following Christ Jesus, something far different from just joining a church or reading its literature. The truth is that we have so much to learn of the actual nature of man through the Master's example that we'll never be in the position of adding to Jesus' teachings. As Mary Baker Eddy explains, a scientific Christian necessarily sits at the feet of Jesus.