Definitions from Webopedia
Interoperability: The ability of software and hardware on different machines from different vendors to share data.
Proprietary: Privately owned and controlled. A proprietary design or technique is one that is owned by a company. It also implies that the company has not divulged specifications that would allow other companies to duplicate the product.
You can see a real world example of interoperability with consumer electronics. A user can buy a Phillips TV, plug it into a Toshiba DVD, add speakers from Bose and play a Sony DVD with a movie made in Hollywood and it all works together.
For an example of a typical proprietary system, consider the imaginary digital radiography system, ProprioRay. It will only work with the ProprioRay sensor plugged into a ProprioRay connector. The image can only be captured and displayed using ProprioRay software and images are stored in a proprietary ProprioRay format.
Manufacturers like proprietary systems. Consumers prefer interoperability. Experience has shown that vendors will attempt to maintain control unless consumers demand interoperability.