Answer
The equation $Q = cm \Delta T$ relates the heat transferred to the temperature change. For the same mass and heat, the temperature change differs if the substances have different specific heat capacity, c.
Physically, differences in specific heat capacity arise because materials absorb and store internal energy in different ways, as stated on page 290. For example, a substance composed of a complicated molecule might have many so-called "degrees of freedom". When heat is added, the molecules will begin vibrating and spinning, leaving less internal energy for their translational KE. So the overall molecules won't be moving as fast, meaning the temperature hasn't gone up that much. Such a substance has a large specific heat capacity.
In other words, temperature measures the average translational KE of molecules, but doesn't count the energy stored in other degrees of freedom.
