head for the hills vs run away

head for the hills and run away both are verbs.

Word	Noun	Adjective	Verb	Adverb
head for the hills	No	No	Yes	No
run away	No	No	Yes	No

As verbs, head for the hills and run away are synonyms defined as:

Other synonyms of head for the hills include break away, bunk, escape, fly the coop, hightail it, lam, run, scarper, scat, take to the woods, turn tail.

head for the hills (verb)	run away (verb)
flee; take to one's heels; cut and run	flee; take to one's heels; cut and run
	escape from the control of