They both mean: "my emotions were strongly affected". I think when "move" is used, it indicates a visible change in outward behaviour. "When I saw the news report about the famine, I was moved to make a donation." "When I heard about the ludicrous comment she had made, I was moved to laughter." "I was moved to tears by the death of my mother." "Touch" seems to be used when the emotions are affected and the change is internal. "I was touched to learn that my colleagues have deep respect for me." "I was touched when I saw children begging in the street." "I am touched by the love of my family." Also....we can be both touched and moved: "I was so touched by the children begging in the street that I was moved to establish a charity to help them."

They are indeed very similar. In that context they give the same meaning but the difference is that saying 'i was moved to tears' is more specific. A person can be deeply touched without actually crying. Also you can say that someone was moved to something else, such as anger. For example 'Seeing him eating my food moved me to anger and I hit him'. Feel free to message me if you want more explanation or examples.

Well I can understand why you can't tell the difference between the two. There is almost no difference, it means almost the same.