Transform, metamorphose, transmute, convert, transmogrify, transfigure can all mean to turn or change one thing into another or a different thing or from one form into another and different form.
In general, the same differences in implications and connotations are observable in the corresponding nouns transformation, metamorphosis, transmutation, conversion, transmogrification, transfiguration .
Transform may imply a mere changing of outward form or appearance or it may imply a basic changing of character, nature, or function.
Metamorphose may add implications not often present in transform such as that of a supernaturally or magically induced change or of a fundamental change in structure and habits that characterizes the development of some forms of animal life or of a transformation specifically induced by chemical or physical agencies.
In more general use the term carries a much stronger implication than transform of an abrupt, startling, or violent change.
Transmute usually suggests a fundamental change, especially one involving a metamorphosis of a lower element or thing into a higher one.
Convert carries a slighter suggestion of change in kind, nature, or structure than the preceding terms but a stronger one of such changes in details or properties as fit something for a given use or function or for a new use or function.
Transmogrify implies a thoroughgoing metamorphosis that is often grotesque, bewildering, or even preposterous.
Transfigure is often interchangeable with transform or metamorphose , but more typically it suggests an exaltation or glorification of the outward appearance.