I'd say
Catherine is classier. It's less common. However, I thing that
Catherine requires an equally classy middle name.
Katherine is more common, and not, in my opinion, as classy.
My personal favorite is
Katharine, mainly because it ties back to my
German heritage, and it is the least common of what I call the Four Main Spellings:
Katherine,
Kathryn,
Catherine, and
Katharine.