(SPOILER)I believe the answer is 50%. For any one segment of the line with n segments, the portion that does not correspond to the same numbered segment in the line with n+1 segments is represented by the equation 1/n - 1/(n+1). For example, suppose n = 2. Segment 1 of each line would take up 1/2 and 1/3 of the length, respectively. The difference in length is 1/2 - 1/3 = 1/6.

However, as it's getting late, I need a bit of time to reflect on what the correct way to go about proving that my answer is correct for all values of n.