Is the motor cortex identical to the sensorimotor cortex?
The primary somatosensory and primary motor cortices are distinct cortical areas. According to the structural classification of Brodmann, the primary somatosensory cortex is referred to as Brodmann's areas 1,2 and 3 (BA1, BA2 and BA3). The primary motor cortex is referred to as Brodmann's area 4 (BA4). The primary sensory cortex is sometimes denoted by S1, and primary motor cortex by M1.


Source: My Brain Notes  

The primary somatosensory cortex is concerned with the processing of somatosensory input, such as tactile stimuli coming from the skin. The motor cortex is concerned with executing movements and hence with motor output. As such, there is a functional as well as structural distinction between the two areas.  

Within the primary somatosensory cortex, basic processing of somatosensory information takes place in BA3, while more advanced processing takes place in BA1 and 2. Neurons in the somatosensory cortex project to adjacent regions, including the motor cortex (Kendal et al., 2000). 

Because the primary sensory and motor cortices are four strips of cortex closely situated together, and because they are functionally coupled at the physiological level, they are sometimes collectively referred to as the sensorimotor cortex. 

Reference
Kendal et al. 2000. Principles of Neural Science