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.