For a culturally different place to spend some time I'd recommend Morocco. I've been both places for extended stays and find Istanbul more like Europe in general than Morocco. Consequently, when I long for a place unlike Europe to combine with a European trip I always now go back to Morocco.

My favorite places there are Tangier/Chefchaouen in the north and Marrakech/Essaouira in the south.

We have been in many parts of the world, included Istanbul. We have really loved Istanbul, but we can not really recommend a wonderful, Cultural, Breathtaking, must seen place better Than Morocco.
I give a vote which 100% assure you to have an amazing time in Morocco better than any country around, we have through Egypt, Istanbul, most parts of Euroup but Morocco was the highlight of our lifetime travels.
Istanbul has some historical sights to see, but Morocco has all: many things to do as activities, and many things to see as must seen places.
go through Marrakech, the High atlas Mountains, Kasbahs esp the UNESCO world heritage Kasbah, Ait-Benhaddou, the Dades and todra Gorges, we loved our desert stay experienced Berber life, camel ride at the golden Erg Chebbi sand dunes, Fes as the History of Morocco.
All of it was amazing as Essaouira as a relaxing place to end up your trip before your departing day.
You will not really regret if you go to Morocco, because it's a worth to see country.

www.weather2travel.com Climate Guides would factor in a lot.

In summer www.goturkey.com Istanbul et al

In winter www.visitmorocco.com

I like Exotic High Atlas Imperial Cities are facinating to me

Happy Deciding really you cannot go wrong with either.

travel.state.gov for a heads up on safety Turkey better there.

Yes, pay attention to the season but not really as simple as "In summer Istanbul...In winter Morocco". Lots of climate variations in Morocco so, yes, pay attention to the season and the particular places you'd like to go.