In the first example that you presented, "fill in the blank with the correct answer" is the idiomatic, most appropriate phrasing. (This is not be confused with to "fill up the blank..." and "fill out the blank...", each of which is wrong usage in this particular case.)
In the second example that you presented, "Write your answer on the line" is inappropriate, since the preposition "on" gives the wrong idea that the answer must be superimposed on the line. The better phrasing is "Write your answer on the space provided" or "Give your answer on the space provided."
I must disagree with the idea that there is not much difference between the usages discussed above. They are semantically and stylistically different and need to be used as the case may be, not interchangeably.