Apparently it differs when you talk about these two words in English and in Danish, then.
But we're using English. In Danish, I bow to your expertise.
In English, 'breathe' and 'respire' are not actual synonyms, although I will grant that they are often used as such - 'respiratory distress' for 'can't breathe', etc. Because of this usage, the OED (which records English as it is commonly used, not as is pedantically correct) lists one as the definition of the other.
Scientifically, which I suspect was what
dane was getting at, this is inaccurate - breathing is actively intaking or exchanging gases to be used in or which are produced as by-products of respiration. Respiration is a chemical process that occurs on a cellular level to oxidise glucose or other 'fuel' molecules and release usable energy. This is more than mere pedantry, the two things
are connected to each other, but are in fact very different.
However, your initial point stands - Ants
breathe in the sense that they actively exchange gases, they just don't breathe like humans.