- meriting condemnation or blame especially as wrong or harmful :
The defendant is culpable for her actions.
- archaic : GUILTY, CRIMINAL (M-W)
So, to use it as a synonym of "guilty" is archaic, and this is the first meaning with which it was first ever used in English, apparently in the 14th century. It is interesting to know its etymology:
Middle English coupable [which is the current French word for "guilty"], from Anglo-French cupable, culpable, from
Latin culpabilis, from culpare - to blame, from culpa - guilt
So there is clearly a connection between culpable and guilty. If you say that something is "culpable", it means that it is liable to blame (as it is suggested by the adjective suffix -able), considered blameworthy by others in accordance with a commonly accepted ethical standard.
Guilt is stronger in that the idea of wrong is more inherent to it ("guilty" does not emphasise the fact that it is "blamable" by others, whereas "culpable" does).
Also, we mostly say to feel guilty, not to feel culpable (as culpable is used from the point of view of those who blame, not from the point of view of the blamed). In fact, "I feel guilty" can have a positive connotation:
I feel so guilty, leaving all this to you. (Collins)
Here "guilty" sounds more like "remorseful", the speaker wishes to assure the other that he/she is not indifferent to the other's difficulty in bearing the burden of "all this". "Culpable" cannot mean "remorseful".