Consumer PR denotes any activity that targets consumers. Corporate PR refers not to the audience, but to the subject, suggesting a focus on an organisation itself (source PR Week).
So, whilst Corporate PR looks at your brand reputation in the long run, Consumer PR focuses on engaging campaigns that can drive brand awareness, and ultimately sales.
Corporate and Consumer PR have therefore different objectives but can sometimes work together especially when you are trying to align your corporate brand with your consumer brand.
Whatever you are working on, make sure you know the key differences in objectives of these two PR approaches.
If you need advice on Consumer and Corporate PR, feel free to get in touch at firstname.lastname@example.org