While I initially interpreted the phrase with a modern understanding of "care", as in to tend to, I did a check on the etymology of "care" to see if it could have possibly have another meaning and this is what I found on Etymonline.com:
care (n.) Old English caru, cearu "sorrow, anxiety, grief," also
"burdens of mind; serious mental attention," from Proto-Germanic *karo (cf. Old Saxon kara "sorrow;" Old High German chara "wail, lament;" Gothic kara "sorrow, trouble, care;" German Karfreitag "Good
Friday"), from PIE root *gar- "cry out, call, scream" (cf. Irish gairm
"shout, cry, call;" see garrulous).
Based on the interpretation of care to mean anxiety, burdens of mind, I believe the 2nd half of the quote is saying that the medicine, though it "cares" for the wound, also brings worry and concern when applied too late.