put down vs lay to rest

put down and lay to rest both are verbs.

Word	Noun	Adjective	Verb	Adverb
put down	No	No	Yes	No
lay to rest	No	No	Yes	No

As verbs, lay to rest is a hyponym of put down; that is, lay to rest is a word with a more specific, narrower meaning than put down:

Other hyponyms of put down include rail, lay, blow, bury, entomb, inhume, inter, rebury.

put down (verb)	lay to rest (verb)
put down in writing; of texts, musical compositions, etc.	place in a grave or tomb
make a record of; set down in permanent form
put in a horizontal position
remove (cargo, people, etc.) from and leave
cause to sit or seat or be in a settled position or place
put (an animal) to death
reduce in worth or character, usually verbally
cause to come to the ground