What does 어 천만에요 mean?

See a translation

감사합니다 and 고맙습니다 are no different in that sense. 고맙다 means 'I am heartwarming and joyful about the help of others', 감사하다 means 'I have a heart of gratitude, a greeting that expresses gratitude.

감사 is Sino-Korean(Chinese character)
고맙 is Pure Korean

Nowadays, there is a tendency to think that 감사하다 tends to be more formal than 고맙다, but it is incorrect/not true. Both 감사합니다 and 감사합니다 can be used. However, if it is possible recommending the expression "고맙습니다", Should use Pure/Native Korean(고맙)

고맙(습니다) is adjective 감사(합니다) is either verb or adjective

감사합니다 & 고맙습니다 are Both polite(more formal)
These are used to olders, elders, seniors, teacher, boss, army, officers, everyone.
(high formality / high politeness)

감사해요 is (a little) informal polite
고마워요 is (more) informal polite

감사해요 & 고마워요 are Both informal (but still) polite
These are used to strangers(someone you don't know), olders, seniors, lover
(low formality / high politeness)

감사 is Sino-Korean(Chinese character)
고맙 is Pure Korean

고마워, ㄳ or ㄱㅅ(Slang which abbreviated the word 감사) is informal or casual used to the same age(friend), someone younger than you, the children, lover, wife & husband
(low formality / low politeness)

고맙다 is formal non-polite(impolite, informal) used to the same age(friend), someone younger than you, the children
(high formality / low politeness)

formality: formal / casual
politeness: polite / rude