Ira and furor are more or less the same as anger and fury in English. The related verbs irascor and furo differ to the same degree. Furor has a more physical aspect : if you like, it can be a physically observable quality in its own right.
In the caeli furor aequinoctialis of Catullus 46.2, ira instead of furor would not suggest to me the actual, physical power of the high winds typical of the vernal equinox. On the other hand ira, anger, is a more of a sentiment, recognizable mainly by its consequences . According to Horace (Ep 1, 2, 62), ira furor brevis est; but I don't think you would say furor ira brevis est.
It's no more than barely true to say that the two words are 'apparently not synonymous'. In English you can be gripped by fury or seized by anger, and you can be either angry or furious. The difference really isn't very much, and the two are often interchangeable, from Latin into English or vice versa. The translation is just a matter of appropriate selection for the context.