I was once asked by a student if Christian Science and Scientology were the same thing. At the time, the idea that someone would confuse the two had ever even occurred to me. The fact that the connection needed to stated in this article on Christian Science reminded me again of the problematic appearance of similarity.
The first problem is obviously the name. There are a lot of Americans who pretty much presume all religions are branches of Christianity. I suppose the thought is "Scientology" is some sort of nickname.
But there are other reasons for the confusion as well. Both religions put forth that their beliefs "when systematically applied to any situation, brings about expected results." Both religions also have a history of shunning certain medical practices, holding their own faith to be more effective and/or legitimate in terms of treatment.
But the two are, in fact, entirely different religions with very little directly connecting them. Which the Church of Scientology states that you do not have to give up your faith in Christianity or other religion, it does not specifically teach about God. Christian Science, on the other hand, is a branch of Christianity. It acknowledges God and Jesus and is focused upon them. It recognizes the Bible as as its holy text.