To "coddle", used in the sense of "to spoil", for example, a child, is a well known word. The other is known to few. I suspect it is used in the phrase "coddled and cosseted" primarily because the alliteration sounds sweet to the ear. Upon hearing the phrase, most people recognize the first word and ignore the second, figuring that 50% is good enough. You have piqued my interest, however. I was among those who had no idea what "cosset" means, so I did some research. The difference seems to be that "coddle" is related to cooking, whereas "cosset" is related to the raising of animals, in particular of lambs and their bedding. Both refer to some sort of gentle treatment, either to the food or to the animal. The net effect is that the expression "coddle and cosset" conveys a sense of taking care of something, probably a living creature, by seeing to its careful nutrition and shelter. However, we cannot be too precise about this. Even the word historians have their doubts about the origins.

Hello! There are no differences between the two words. Sometimes in english, expressions may use similar words within the same sentence to add depth to their explanations. “The woman grew with joy and happiness.” “The child threw a tantrum out of rage and anger.” I hope this helps. Cheers!