More generally, I believe you're looking for a quantitative measure of similarity between functions. This is a large domain of research. The best measure will always depend on the problem and signals at hand, so there is no simple answer. I can recommend the following:
- Take a look at this survey paper: Comprehensive Survey on Distance/Similarity Measures between Probability Density Functions
- I once was working on quantifying the similarity between spiky signals (i.e., signals with fast decay), the normalized mutual information was the best measure for me.
- Experiment with several measures, until you find the one satisfactory for your application.