In this specific case isreal asked residents of Gaza city, a subpart of the Gaza strip to move to a more southern position in the Gaza strip. Like asking people to move from Manhattan to Brooklyn because they are going to blow up Manhattan. People are capable of moving within the Gaza strip. That doesn’t mean there will be housing for them however.
I have not used either for cars but I have tried both for driving bicycles. The thing with open street map is that they have a lot more of the small streets that bikes can go on but cars can’t. In my experience google maps usually wants me to drive my bike right on some huge multi-lane car road for 3 hours and totally ignores the bike path right next to it. When I went on a 1 month bicycling trip I tried both but found that OSMAnd had vastly better suited roads. Some of the tiniest tiny roads it sent me on were some of the most memorable of the whole journey. Sometimes the path in OSMAnd will just be a dirt path half a meter wide and I love that. I thought many times who on earth even added all these roads! I’m so thankful for every local nerd who added every single dirt road in his neighborhood on there. Simply amazing sometimes. But yeah it doesn’t work at all if you don’t download the map that’s true. Also I really can’t answer for navigating by car.