Which correlation coefficient is the most appropriate to compare 2 time series? I want to compare the variation of one variable for 2 regions, have regional data for the last 30 years. Is Pearson correlation ok or should I rely on Kendall's tau b or Spearman's rho and why? I tried to google it and analyse what I found, but I'm still not sure.
For time series some version of Pearson correlation is most used, in the form of the autocorrelation function (for one series, correlated with itself at various lags) and the cross-correlation function (for two series) likewise. They are correct when all conditional expectation are linear.
If you suspect that may not be the case, you should start with some visualization of the two series! I have not seen any detailed descriptive analysis of two time series, that would be rather interesting ... In R you could play with the function
coplot and you could make scatterplot matrices, replacing what would be one number in each of the two functions above (autocorrelation, crosscorrelation) with a scatterplot. You could also look into copulas used with time series.