Well, as far as I understand it, both 감사합니다 and 고맙습니다 are polite, but 고맙습니다 is a bit more friendly sounding. 고마워 / 고마워요 is also another way to say "thank you". It's usually used between friends, I believe. Hopefully a native speaker will answer your question / correct my answer as needed.^^

Sarah is right. Just remember that you can say 감사해요, 감사합니다. And you can't say 감사해. Because 감사해요 and 감사합니다 are polite expression only.

I think both are the same and have got almost the same politeness, but I recommend "고맙습니다(谢谢)" instead of "감사합니다(感谢)" because "감사합니다" comes from the Japanese word "感謝" while "고맙습니다" is the pure Korean.:) As for "감사(noun)", In Korean, "감사(感謝)" - gam sa (we also use 繁体 like Hong kong,Taiwan) In Chinese, "感谢" - gan xie In Japanese, "感謝(かんしや)" - kan sha How interesting they are..:) As far as I know, there are no other words to express the same meaning with them, but both expressions can be conjugated in various ways. As Sarah and Wonner mentioned, "고마+워" (between friends) "고마+워+요"(a bit polite) "고맙습니다"(very polite) In any cases, "감사합니다" is not used between friends. "감사+해" (X) "감사+해요" - acceptable but seldom used in conversations.

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