완치 完[완전할 완] 治[치료할 치] - 병이 완전히 낫다 쾌차 快[나을 쾌] 差[나을 차] - 병이 완전히 낫다 완쾌 完[완전할 완] 快[나을 쾌] - 병이 완전히 낫다 The origin of the words are different. But the meaning is same.

although word-for-word the three means the same literally, nuance and actual usage in South Korea differs some-what: (deep-rooted meanings in each Chinese character do hint) : 완쾌 better used when you are well again from light symptoms. 완치 when you have recovered fully from serious ailment, with long-term treatment and/or hospitalization. 쾌차 looks more towards the future for a (re)fresh(ened) start: from now-on, after what you have gone through. : however, in Post-Millennium South Korea amongst chat, tok and blogs: all three above are mostly used on elders and official correspondence, as younger generation simply get by all of the above circumstances with "다 나았니?" albeit: are you fully-recovered now? : ( again, in the mass-aging population: "younger" above would mean below 30's)

완치 is more likely to used to say "요즘에는 완치될 수 있는 암의 종류가 꽤 있다 (Many cancers can be cured these days" or like that then when you hoping somebody gets well. I don't see any differences between 쾌차하시길 바래요 and 완쾌되시길 바래요 but I would say 완쾌되시길 바래요. I'm more familiar with that expression. This is just my personally opinion.