The external rigid insulation is better for comfort and energy performance in every way, it just requires careful installation of flashing around doors and windows.
I didn't think about the extra steps I might need for the flashing around door and windows.
You have to provide drainage for any water that condenses or gets behind siding.
That is why the siding is not nailed straight through the foam (touching it) but is instead nailed to the small wood peices leaving a 1/2 or more air gap between the siding and the foam to allow hot air in the summer and moisture in the winter to run down the foam to the dirt.
If you install a lot of rigid say 2" or more you might be looking at having to install strapping which adds time and expense to the installation.
What do you mean "strapping"?