"Significantly" means that something is not just surprising, but that it means something. You expect the writer to tell you next what it means. "Rutherford's team discovered that a few alpha particles were, significantly, bouncing straight back from the gold foil. Rutherford realized that this meant that Thomson's model of the atom must be wrong." The word is related to "significance" which is a synonym for "meaning."
"Remarkably" means that something is amazing or surprising. "He wasn't just tall, he was remarkably tall: 200 cm." "Remarkably" means it is so surprising that you are likely to say something about. You are likely to make a remark.
"Apparently" means that it looks that way, but you aren't quite sure. "I don't see any lights on in their house. Apparently they aren't home... unless they have gone to bed very early." "Apparently" means "it appears that way."