Why liquid in a duct flows when there is a pressure difference?

No. When a pressure difference exists between two points, the fluid will start to move from the higher pressure point to the low pressure point. This happens to minimize the pressure difference.

Without such a pressure differential, the fluid is stationary, and the system is absent of any flow.

But if we consider a long horizontal pipe, and consider a small segment of this pipe, Bernoulli's equation tells us that since $$p_1 +rho gh_1 + frac{1}{2}rho v_1^2 = p_2 +rho gh_2 + frac{1}{2}rho v_2^2$$ then since $h_1=h_2$ $$p_1 + frac{1}{2}rho v_1^2 = p_2 + frac{1}{2}rho v_2^2$$ Now since we are considering a small section, we get that $p_1-p_2approx 0$ which will yield $$v_1=v_2$$ meaning that we can have fluid flow over a small pressure differential.

But again, for large sections, the pressure at one end of the pipe must be larger than the pressure at the other end for there to be flow.

Considering the motion of a perfect fluid is like considering a mass that slides without friction: it can remain at constant speed indefinitely without any external force. It is the same with a perfect fluid that can flow indefinitely without a pressure differential. In a sense, it is what happens for a superfluid of zero viscosity (But the subject is complicated!).

When there is no potential energy term (same height) Bernoulli's theorem simply tells us that, for a perfect fluid, a pressure difference causes a change in speed. This is equivalent to Newton's law: a force produces a change in speed. In contrast, for a viscous fluid, the pressure differential compensates the viscous forces and maintains a constant velocity.