What is the difference between 감사합니다 and 고마워 and 고맙다?
Feel free to just provide example sentences.

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