The distinction between precision and accuracy is one of the most useful distinctions I've learnt.

If your goal is to get at the truth, then accuracy is always the primary goal, and precision secondary. Indeed, it is quite dangerous to aim for precision first. This was also captured by Knuth, "premature optimization is the root of all evil."

Unfortunately, most people are convinced more easily by precision than by accuracy. Politicians and false prophets often employ this trick. Precision reflects confidence. Also, it is trivial to very whether a statement is precise; but incredibly difficult to verify if it is accurate.