The central differentiating factor is that Sephardic Haredim follow the rulings and customs of their rabbis, which are rooted in the traditions of the Jews formerly from the Islamic world.
Porat Yosef Yeshiva quickly became a reputable Sephardic institution, educating many students and even "exporting" Rabbinic leadership to Jewish communities in the Muslim world.
Alongside Porat Yosef Yeshiva, there were a number of Sephardic Haredi Rabbis, not affiliated with any particular institution nor with each other, who were strongly opposed to Zionism, the State of Israel, modernity, and secularism, much like the Askhenazi Edah HaChareidis.
Their attempt to form a Sephardic counterpart to the Edah, HaEdah HaChareidit HaSpharadit, was largely unsuccessful, as it failed to attract a significant membership.
During this period, there was wave of Sephardic and Mizrahi Jews who were educated in Lita'i Haredi Yeshivas, and consequently adopted the worldview and lifestyle of the Ashkenazi Haredim.