A communication designer uses visual elements to deliver specific information to the viewer. It is more about forming a relationship with your audience and the visual. It is their job to strategically communicate the brand's messages to their target audience.
Communication design is a broader term and it refers to using graphic design elements to strategically deliver a message. Although the jobs look similar, the roles are different. A graphic designer might create the graphics using software, but it's the communication designer's tole to choose the phrases/messages and mix them with the graphics to create a visual. They also need to have the skills to organize design elements to create an appealing design.