Welcome Dan! Love your choices.
I have only researched the Portuguese, not walked it . Yet . (It's high on my list, but I want to walk it in its entirety).
Statistically, the temperatures seem to vary - sometimes the Primitivo is warmer, sometimes the Portuguese. I believe overall it's cooler on the Primitivo.
But in the unsettled world we now live in, it's anyone's guess...
I've just walked the Primitivo, and when many were sweltering, I had high's of just 20 - 25 degrees, beautiful hiking weather. Mornings were always cooler.
Mid - late July, it was definitely warmer on the Portuguese. ( A friend was on the trail).
Ironically, today the Primitivo is warmer than the Portuguese! By around 3 -5 degrees - the difference between comfortable and getting a little warm. So in this respect it's a hard call.
Number wise, judging from the statistics, the Portuguese has significantly more pilgrims. I'm decidedly socialable, but I also love walking alone - the Primitivo gave me the space to be both, as I wished. If you're wife is more socially inclined than yourself, it's a great option!
If you're covering fairly normal stages, you'll complete it in roughly 14 days, leaving you either time to sightsee before/ during/ after the walk, ( think recovery from possible jetlag, enjoying a rest day in Lugo, and a couple of days in Santiago) . Plus a day or two back home before you have to return to work.
(I'm assuming your three weeks is going to include travel time )
Whichever you choose, Buen Camino!