The answer is not to bury the fence posts in the earth or concrete - keep them on top. There are many systems like: Simpson Strong Tie E-Z Base.

Google 'mount fence post to concrete'.

Barn poles are going to be much deeper and depth does the holding. Concrete on fence posts adds rigid mass that is less prone to movement as compared to soil alone, which is glacially fluid. The two possibilities are matters of DEGREE, not of absolute right and wrong. Concrete is going to be more solid. Some fences are heavier or have more wind resistance and could use a more solid anchor. It depends on all the variables as to what makes more sense for a given project.

Posts should be set for drainage. Ever see a heaved railroad bed? Me neither.