Happen vs. Occur

The formal definitions of both words are similar.
There are some differences, however:

Happen — something random to take place;
Occur — something to become observable; in a certain moment; an event or a process;

The difference is even more evident if you notice that oc cur is relative to current as they both came from Latin currere "to run".

So, I would use occur in the following contexts:

Formal conversations;
When a time or place of an event is specific; Expected events, like dates: The Spring Bank Holiday that would usually have occurred at the end of May;
Discoveries, ideas: The discovery
occurred in Mono Lake, California ;

...and happen in the following contexts:
Informal context;
Rough time or place;
Random events, usually without being planned!

Thank you