highness vs degree

highness and degree both are nouns.

Word	Noun	Adjective	Verb	Adverb
highness	Yes	No	No	No
degree	Yes	No	No	No

As nouns, degree is a hypernym of highness; that is, degree is a word with a broader meaning than highness:

Other hypernyms of highness include grade, level.

highness (noun)	degree (noun)
the quality of being high or lofty	a position on a scale of intensity or amount or quality
a high degree (of amount or force etc.)	the seriousness of something (e.g., a burn or crime)
	the highest power of a term or variable
	an award conferred by a college or university signifying that the recipient has satisfactorily completed a course of study
	a measure for arcs and angles
	a unit of temperature on a specified scale
	a specific identifiable position in a continuum or series or especially in a process