고맙다 vs 감사하다

Q. What's the difference between '고맙다'and '감사하다'?

A. '고맙다' is an adjective whereas '감사하다' is an adjective and a verb at the same time. 

So only '감사하다' can combine with verb conjugations. 

For example,

항상 감사하고 있습니다.
항상 고맙고 있어요(X)
-> 항상 고마워하고 있어요.

Two words are interchangeable when they are adjectives according to the dictionary.

However it is subtly different in real usage.

Koreans use '고맙다' when they say something to someone similar in age or younger.

친구야~ 고마워.
친구야~ 감사해(?)

To older people, they use '감사하다' more often.

선생님,  감사합니다!

It also depends on the situation.

People prefer to use '고맙다' in casual conversations whereas they use '감사하다' in formal situations like a business relationship. So '감사하다' is used with formal endings.

언니,  고맙습니다(?)
-> 언니, 고마워요.
사장님, 감사해요(?)
-> 사장님, 감사합니다. 

I think '고맙다' is more like 'thank' and '감사하다' is more like 'appreiciate' in English. 

Sometimes you are in ambiguous situations or those kinds of relationships.

If there is a colleague who has a similar age as you but not close to you, which word will you choose?

You can say both in many different ways like you can '고맙다' with formal endings or '감사하다' with casual expressions. 

지민 씨,  고맙습니다.
지민 씨, 감사~

I hope you can find many different usages in your life experiences. Gook luck!