"Precision", from its Latin roots, originates with the notion of the removal of all inessential elements: not too much, not too little. Just right.
"Exact", from its Latin roots, originates with the notion of highly finished performance.
"Accurate", from its Latin roots, originates with the notion of caring, or "done with care".
All the words retain the emphasis presented by their roots. Here are some examples:
Suppose you seek a Phillips screwdriver to fit a certain screw. If you find one that fits precisely, it means it fits the screw with neither too much nor too little extra wiggle. That would be a "precise" fit.
You could also say the fit was "exact", but that word works better to describe a job well done. For example, you could say a singer sang a song "exactly" right, but "precise" and "accurate" would not seem like artistic choices.
A student who gives correct answers to all the questions on an exam would be said to have given "accurate" answers, not "precise" or "exact" ones. He took the care to do it right.