In reply to Sunmee.

I do not know how to do "a statistical test for the differences" between two arbitrary curves. For linear curves, you can ask whether the slopes and intercepts are different. For a general nonlinear parametric curve, you might be able to test whether the parameters are the significantly different. But I don't think your question makes sense for a LOESS curve, which is nonparametric. You could estimate the Root-MSE between two curves, but I don't know how to use the result in a statistical test for the difference. Statistical tests for a null hypothesis are based on knowing the distribution of the sampling distribution of the test statistic under the null.