The Online Etymology Dictionary gives this description for the etymology of boron:
Originally called boracium by Humphrey Davy because it was drawn from boracic acid.
So that form was originally used by the discoverer of boron. I think it is fair to assume that borium is just a simple Latinization of boron.
Azote was the original name coined for the element nitrogen, so its use is similar to boracium for boron, though additionally it is used as the term for the element in some other languages (see Wikipedia). Nitrogenium is also presumably just a Latinization of the English name.
So, at least for these two, it seems to be a preference for an older name for the element (though not a classical Latin one) versus Latinization of the modern English name.