"No, thank you."
감사함니다 is more formal. Theyre both formal versions of thank you, but think of the latter as a way to thank someome much higher ranked than you a stranger, or someone much older than you.
고맙습니다 is native Korean and 감사합니다 is inherited. They're both formal but 고맙습니다 is a bit more friendly sounding :)
Is in not supposed to be 아니요, 고마워요? Or is this particular "no thank you" context based?