Aside from translation differences, I was never sure of what distinguishes them from each other. Anyone else know?
They are basically the same. They are many set phrases to celebrate a happy new year, and your examples are only some of the most popular ones. But all-kanji versions would look stiffer, and they are meant to be written/printed on 年賀状. In speech, you can stick to 明けましておめでとうございます everywhere.