Jan vs January

Jan and January both are nouns.

Word	Noun	Adjective	Verb	Adverb
Jan	Yes	No	No	No
January	Yes	No	No	No

As nouns, Jan and January are synonyms defined as:

Jan and January: the first month of the year; begins 10 days after the winter solstice

Jan (noun)	January (noun)
the first month of the year; begins 10 days after the winter solstice	the first month of the year; begins 10 days after the winter solstice