The topping check is a catch-all for lots of possible checks. If you are simply making sure that you can squeeze out a max N1 or fuel flow, it is simply a check that you can get to maximum power, without necessarily checking what that max is.
If you compare measured N1, temp, or torque at an ambient pressure altitude and OAT against a chart for the minimum spec engine, then you are doing a one point power check, and measuring the engine against the standard one used to calculate the performance of the aircraft. This topping power check also proves the other topping check at the same time.
A part power "power assurance" check is used to measure the engine against a standard one at low power (which saves wear on the engine). This confirms that nothing bad has happened, and that the engine should still produce full rated power when called. If the engine type can have less margin at high power than at lower power, the power assurance check may have a minimum power margin as called out. This is a clue that the engine might naturally have lower margin at full power, so the extra margin is tacked on to the lower power.
What goes wrong is usually that the engine gets dirty, eroded, fodded, or a bleed leak so it can't produce the power it should, and maintenance must be performed. Also, if the pilot choses to do the check with a crosswind or downwind, some exhaust might be sucked in that spoils the readings by heating up the inlet air and making the power drop.