【Improve】 ,  【recover】 ,  【recuperate】 ,  【convalesce】 ,  【gain】  are comparable as intransitive verbs with the meaning to grow or become better (as in health or well-being).

【Improve】 , although often employed in respect to health, is also applicable to situations or conditions and indirectly to persons. In reference to health  【improve】 implies nothing more than a getting better; it connotes hope but no certainty of continued progress or of final achievement of full health.

【Recover】 usually implies a return to or the regaining of some former or normal state (as of health); the word may, especially with reference to health, imply certainty and not merely hope.

【Recuperate】  comes very close to  【recover】 in its implication of getting back what has been lost and is perhaps more common in reference to losses of money or energy. In respect to health it especially implies restoration through such influences as climate and rest.

【Convalesce】 fundamentally implies a growing stronger; the term usually applies to the period between the subsidence of a confining illness and full recovery, when the patient, more or less gradually, gathers strength and regains the use of powers lost or depleted through serious illness, a serious operation, or a serious wound.

【Gain】  simply means to make progress especially, but not always, in health. The term is used typically in periodic reports of condition and like  【improve】 carries no implication of whether or not progress will continue or result in permanent recovery.