One holds that the establishment of the State of Israel is required by halacha and is the beginning of the Geula, and it is a mitzva to support the Medinah, and that the future of the Jewish people is tied up with the survival of the Medinah.
The other holds that the establishment of the State of Israel was not only prohibited by Torah, but is also a dumb idea, and that the zionists are similar to Shabbatai Zvi and the Frankists, and no good end will come of this, and the survival of the Jewish people will be in spite of the Medinah.
Other than that, they get along fine. They read each others sefarim. They often work together on non-political matters. Many people with hareidi views are quite modern, and many of the religious zionists are quite frum. The leading rabbanim of both camps are cousins.And most baal ha-battim are somewhere in the middle.