watts or joules?

In those applications there is little to no difference -- just the way it's expressed.

Joule is a unit of energy (i.e. it takes 1 joule of energy to exercise the force of 1 newton for a displacement of 1 metre).

In electronics it related to Watts as follows:
1 joule= the equivalent of 1W of POWER per second required to produce 1Watt in one Second. Similarly, Watt is the product of amperes times volts (voltamperes).

Basically, you can look at a power supply's watt rating and be just as well informed. Much of the stuff you see is mumbo-jumboed to sound more scientific and different, and, thereby, more luring than it actually is.

A joule is one watt multiplied by one second (watt-sec), or energy consumption over a period of time. This is what your utility company bills (a kilowatt-hr is 3,600,000 joules). Or, the units of joules is voltage x ampere x time.

The different joule rating of the amps reflects the amount of time required to deliver a given current (based on the speaker impedance only) at a constant voltage. To oversimplify: if volts times amps is constant, the difference is in current delivery time. This is determined mostly by the power supply capacitance.

This time difference is the equivalent of having power in reserve for transient spikes that exceed the steady power rating of the amplifier. The capacitors trade voltage for current and discharge the energy as the speaker calls for a transient spike. If the caps have the joule rating to do it, the spike is transduced and you hear a nice attack on the instruments played.

The power that the speaker draws to produce sound is determined only by the amplifier output voltage and the speaker's impedance. But this is steady power - constantly available at a given impedance. A variation in the speaker impedance causes the current demand to change. A lower by half impedance at a certain frequency will cause the current draw from the amp to double, if voltage remains constant. This extra current has to be available either from the power supply capacitors or transformer. It it's the transformer that's capable, the higher current will flow all day long (the joule rating is infinite since the joules are transferred as long as the amp is playing). But if the extra energy is stored in the capacitors, the joule rating dermines how much and by how long you can have the extra current for. This is basically the headroom that an amp has.

The bottom line is the 360 joule amp will perform better if and only if the power rating and transformer VA ratings are identical, all other things being equal - which is rarely the case.