I'll argue that you cannot assign any statistical significance to the difference between two sets of points as curves, when you have ONLY a list of points but no model. That means you have no direct way to infer the noise structure in those curves. And when you ask if two curves are "significantly different" that is exactly what you are trying to do.
You don't say if the curves have the same x-values, but different y, or if the curves are sampled at disparate points on the x axis. You don't even say if the curves live on the same support. (The same range of x-values.) This problem could become quite difficult, depending on how you respond on those issues.
If the curves are sampled over the same support, at identical locations, then you might be able do something statistically valid, treating the points as replicates.
Another idea might be to approximate each curve using smoothing splines. That will give you a simple measure of the noise structure in each set of points from the residual errors. Then you might be able to come up with a test that measures how statistically different are the curves. It would take some serious thought though.
