Omeka.net runs on Omeka Classic. It’s useful if your institution’s hosting doesn’t meet the system requirements for Omeka Classic, and/or if you prefer to have all of the plugins and themes pre-installed (rather than dealing with ftp and server files).
There is some discussion of Omeka S vs Omeka Classic in the FAQ here on the forum.
For real-world examples, check out the showcases for Omeka Classic and Omeka.net, and the directory of Omeka S sites.
For oral histories, you might want to take a look at the Oral History Metadata Synchronizer plugin suite for Omeka Classic. You can see it in action on Goin’ North, which is linked from the OHMS page.