When the pulley has mass (is not "light"), you must account for its rotational inertia.

Atwood machine with massive pulley (mass M, radius R): For masses m1 > m2 hanging on either side:

• For m1: m1g - T1 = m1a
• For m2: T2 - m2g = m2a
• For pulley: (T1 - T2)R = Ialpha = ($MR^2$/2)*$\frac{a}{R}$ => T1 - T2 = Ma/2

Adding: a = (m1 - m2)*g / (m1 + m2 + M/2)

Key point: T1 ≠ T2 when the pulley has mass. The tension difference provides the net torque on the pulley.