handsome vs better-looking

handsome and better-looking both are adjectives.

Word	Noun	Adjective	Verb	Adverb
handsome	No	Yes	No	No
better-looking	No	Yes	No	No

As adjectives, handsome and better-looking are synonyms defined as:

handsome and better-looking: pleasing in appearance especially by reason of conformity to ideals of form and proportion

Other synonyms of handsome include fine-looking, good-looking, on fleek, well-favored, well-favoured.

handsome (adjective)	better-looking (adjective)
pleasing in appearance especially by reason of conformity to ideals of form and proportion	pleasing in appearance especially by reason of conformity to ideals of form and proportion
given or giving freely