Find My Mac doesnt work if you use an ethernet cable for?
Find My Mac needs two things to work:


An working internet connection, so that the Mac can contact Apple's servers and register its location.
A means of determining the Mac's location.


For the first condition, either a wired ethernet or Wi-Fi connection will suffice. For the second condition, only Wi-Fi will work, because the Wi-Fi radio is used to triangulate a position, based on Apple's database of Wi-Fi networks around the world. Essentially, you can take the names of Wi-Fi networks around you and their relative signal strengths, and Apple's database (and others like it) can determine where you are, within a relatively small radius.

While you can get a rough geolocation based on an IP address, it's not nearly precise enough to be useful for locating a Mac, hence why an ethernet connection alone is not enough (it could probably determine the city the Mac is in, but nothing more concrete).

A few useful things to keep in mind based on this:


A Mac can be located if the Wi-Fi adapter is turned on, but not connected to any networks and the ethernet connection has an internet connection. You can test this easily enough yourself.
If your Wi-Fi is on, but not connected (nor is the ethernet) to the internet, Find My Mac still won't work.
Turning off Wi-Fi will disable Find My Mac, regardless of whether there's any ethernet connection. However the Find My Mac service does cache recent locations (although I'm not sure for how long).


So, for the best chance at recovering a lost Mac, make sure that the Wi-Fi adapter hasn't been turned off. If you need to do so (i.e. getting on a plane, etc.), be sure to turn it back on when possible.