Hello. :)
I'm a native Korean.
1. 묵직하다.
When I lift something which is not shown (for example, it's placed in the bag), I guess something in the bag is heavy. In this case, I say '묵직하다.'
2. 무겁다.
Regardless something is shown or not, if I feel it heavy, I say '무겁다'. '무겁다' is more generally used than '묵직하다'.
Hope it helps :)