According to the lexicons they both may mean to begin, my feeling is however that καταρχω is more emphatic, like to begin from the very beginning
Well, that is a very good place to start… 
They appear to me to be close synonyms, but I don’t think it’s that sort of emphasis, rather to a native speaker the word would sound better or fit better in some contexts than others. For example, LSJ notes “2. in religious sense, begin the sacrificial ceremonies…” a usage which the simplex ἄρχω doesn’t seem to share.