What does the cog wheel in Ubuntu's Settings > Search do?

The cogwheel lets you customise the search locations (for Files).

But why doesn't this work on Ubuntu 18.04?

Because Ubuntu 18.04 doesn't have tracker pre-installed.

tracker is a metadata database, indexer and search tool for GNOME. It increases the speed of searching in the Files app and also supports full-text search. But this comes with a huge cost, it causes high CPU use and/or other performance issues.

That's why Ubuntu developers decided to ship Ubuntu 17.10 and 18.04 without tracker (see known issues).

If you manually install tracker and re-login, the cogwheel should be activated.