If the question is when you should use implicit flat (and often improper) priors and when you should use informative priors, then the answer is that you should typically be using somewhat informative priors. There is a wiki page about priors here:

When you say that it “doesn’t change the estimation accuracy”, you have to specify the estimation accuracy of what? Somewhat informative priors will usually reduce the variance of all estimates, even if they don’t affect the mean or median much. If you are really in a situation where the data is such that the priors are not making any difference, then your model is probably too simplistic and there is probably a lot more heterogeneity in the data generating process that you should be trying to model.