Typically, in the U.S., thyroxine is the natural hormone our thyroids produce, and levothyroxine is the generic name for the synthetic meds. Both are the same thing - T4. However, I believe the terms are used interchangeably by many manufacturers.
Each manufacturer of levo uses their own fillers (inactive ingredients) in their tablets. Some people are allergic or sensitive to some of those fillers; some of the fillers impact absorption of the meds for some people. For that reason, we always suggest that you make as few changes as possible once you find one that works well for you. Consistency is very important.