While I'm not sure of any differentiation between "other" and "the other", I would say that the difference between "another" and "other" is the difference between "additional", and an exclusive choice.
I've had one dessert but want another. (an additional dessert)
You can have this for dessert or the other choice. (the choice is between the two, both is not an option)
EDITED for the updated question: In the sentance provided, there is an omitted but implied bit. Read "one part of speech for another" as meaning "one part of speech for another part of speech", implying parts are being interchanged. So, to go with my original thought, you have one part of speech and this part of speech is also being used in an additional part.
If you were to flesh out that sentence for more clarity, read it as:
In the early part of the modern English period the vocabulary was enlarged by the widespread use of one part of speech for another part of speech, and also by increased borrowings from other languages.