It looks like the 'Industrial Internet Consortium' have simply shortened the name from something slightly more self-explanatory: Industrial Internet of Things. Note in their footer:
The Industrial Internet Consortium is the world’s leading organization transforming business and society by accelerating the Industrial Internet of Things (IIoT).
In a nutshell, IIoT is just the use of 'smart' devices, sensors and machinery in industry, rather than in the home environment for consumers.
For a real-world, tangible example, take Rolls-Royce, who, among other things, produce aircraft engines. It was announced relatively recently that they were equipping their latest engines with sensors in order to monitor performance remotely and predict engine faults before they happen. This is done by transmitting sensor data to Azure IoT Suite and processing it from there, in order to make business decisions without having to physically check the engines.
For a specific definition, I find TechTarget's article pretty useful:
The Industrial Internet of Things (IIoT) is the use of Internet of Things (IoT) technologies in manufacturing.
Also known as the Industrial Internet, IIoT incorporates machine learning and big data technology, harnessing the sensor data, machine-to-machine (M2M) communication and automation technologies that have existed in industrial settings for years.
An article on Electronic Design discusses the differences between consumer, commercial and industrial IoT. The key point they raise is:
IoT systems and platforms are not created equal. The types of communication and operations used by consumer, commercial, and industrial IoT are very similar if not identical. The differences concern those who procure, and are allowed to procure, information or control within the system.
In brief: The Industrial Internet is just IoT concepts applied specifically to industry and manufacturing rather than any other application. This can, of course, bring benefits in many situations to business (although, as with much of IoT, it's important to actually evaluate whether connecting everything to the Internet is a good idea, or even useful at all!)