We investigate various methods for testing whether two groups of curves are statistically significantly different, with the motivation to apply the techniques to the analysis of data arising from designed experiments. We propose a set of tests based on pairwise differences between individual curves. Our objective is to compare the power and robustness of a variety of tests, including a collection of permutation tests, a test based on the functional principal components scores, the adaptive Neyman test and the functional F test. We illustrate the application of these tests in the context of a designed 2^4 factorial experiment with a case study using data provided by NASA. We apply the methods for comparing curves to this factorial data by dividing the data into two groups by each effect (A, B, . . . , ABCD) in turn. We carry out a large simulation study investigating the power of the tests in detecting contamination, location, and shift effects on unimodal and monotone curves. We conclude that the permutation test using the mean of the pairwise differences in L1 norm has the best overall power performance and is a robust test statistic applicable in a wide variety of situations. The advantage of using a permutation test is that it is an exact, distribution-free test that performs well overall when applied to functional data. This test may be extended to more than two groups by constructing test statistics based on averages of pairwise differences between curves from the different groups and, as such, is an important building-block for larger experiments and more complex designs.