If I want to compare two sets of measurements, i.e., how much their means differ or how much the sets differ, I would say I have two options:
Paired difference t-test: Calculate the differences to get the change score for each element and then use the set of differences to build the t statistic (with the mean and standard deviation of the differences) and compare with the t-distribution. The degrees of freedom would be n-1.
Similar approach, but calculating means and standard deviation separately for both samples. Build the the statistic using the two means and standard deviations. The degrees of freedom would be 2n-2.
What are the differences between these two approaches?