Heartstaff is a rare/unique instance of psi-weapon. 15% refers only to the bonus, the base accuracy (of psi-weapons) depends on PST * PSK value of your soldier/attacker, as in
PSATK = 0.02 * PST * PST + RNG[0..55] + bonus (15% for heartstaff) - distance
Enter battlescape, select psi-action and hold alt to view approximate psi-attack (less by 30, because assumes 30 psi-def).
It also depends on psi-defence of your target, which is commonly (for most):
PSDEF = PST + 0.4 PSK + 0.33 Morale + 0.2 Bravery
If your PSATK > target's PSDEF, the attack works.
The takeaway is 1) staff is almost as likely to work as psi-amp 2) both should only really be used by guys with at least high (90/60+ PST/PSK) against guys with low PST/PSK (e.g. hybrid assaults, mibs). Meaning, you need first train (and max out) your troops in psi-lab to use it.
As for where it's useful. It ignores armor and can strike without LoS, so probably some heavy target / multi-level assaults. It's one of the best weapons for dreamscape that you can happen upon later. It should be useful underwater, but aquatoids are resistant (more accurately, have high PST) and I don't remember if it's effective or not - probably you need really strong psionics.