The amount of power (watts) going into your speaker depends on characteristics of your speaker, and how loud you play the music. A 600 watt amp will deliver only what the speaker demands, not what the amp is capable of delivering. So, the 120 watt amp and the 600 watt amp are probably both delivering something like 10 watts rms most of the time. However, the more powerful amp can swing a higher voltage, which if it persisted, would corespond to more watts. But, for music, such high voltages are brief peaks of the music signal waveform, and don't represent much power. The high powered amp will deal with these peaks better than the low powered amp.
So long as the signal is music a high powered amp will not damage speakers. However, if there is a malfunction (IC ground problem perhaps) the high powered amp can toast your speakers quicker than a low powered amp.
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