Should I use Kendall correlation or Pearson correlation?

I am analyzing some data for my research in preparation for a poster presentation this December. I have calculated correlation matrices for the data using the cor() function in R. I have used the Pearson method (which is default) and the Kendall method. I then exported these correlation matrices and plotted the row that I’m interested in correlating. The plots are significantly different.

If I split the data into two classes, the plots look more similar. But if I combine the data into one big set and perform the correlations, the Kendall plot shows more positive correlation while the Pearson plot shows more negative correlation.

In other words, they seem to be showing different results. I’m not a trained statistician, but I’m trying to validate the conclusions I am drawing from this data. I don’t want to choose a statistical method just because it suits my objectives better.

Is there a good reason to choose either Pearson or Kendall’s method for correlation?