degree vs magnitude

degree and magnitude both are nouns.

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

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

degree (noun)	magnitude (noun)
a position on a scale of intensity or amount or quality	the property of relative size or extent (whether large or small)
the seriousness of something (e.g., a burn or crime)	relative importance
the highest power of a term or variable	a number assigned to the ratio of two quantities; two quantities are of the same order of magnitude if one is less than 10 times as large as the other; the number of magnitudes that the quantities differ is specified to within a power of 10
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