In the dictionary,
cube something: to cut food into cubes
dice something: to cut (food) into small cubes
I guess when you "cube something", the cubed cubes can be big or small
but when you "dice something", the cubed cubes are normally small
But I might be wrong.
Oxford dictionary also says "to cube" and "to dice" are the same or synonyms.
Do we care about the size when saying "I cubed the meat" and "I diced the meat"?