ADEN, Yemen — Less than an hour’s drive outside this dilapidated port town, the Yemeni government’s authority is scarcely visible, and a different flag appears, that of the old independent state of South Yemen. The flags are one sign of a rapidly spreading protest movement across the south that now threatens to turn into a violent insurgency if its demands are not met. That could further destabilize Yemen, already the poorest and one of the most troubled countries in the Arab world, and create a broader haven for Al Qaeda here. The movement’s leaders say the Yemeni government — based in the north — has systematically discriminated against the south,...