Assuage defined: to make better or lessen; fulfill the needs of; to pacify VERB —Your Dictionary.com
Some definitions equate assuage with the ministering of a nurse. Assuage would best describe a caring act. The kindness accompanied by the relieving or the quieting of a problem. So to relieve someone or to quiet their ills is to assuage them.
Relieve and quiet used this way bring to mind the tender attentions of a parent, perhaps. As synonyms of assuage, they seem to convey the kindness and caring of that word as opposed to a word that means simply to stop; stopping the pain is not to assuage the pain.
Relieve defined: to reduce or remove (something, such as pain or an unpleasant feeling) : to make (a problem) less serious VERB–Merriam-Webster.com