Interfacing a melody generator IC that drives speaker directly to an amplifier
The problem is probably common mode noise.  Since the chip is driving both ends of the speaker, it is free to add arbitrary common mode signal on top of the differential audio signal.  That common mode signal would mess up attempts to filter and use either of the outputs individually.

Given what the chip does and how it works, the best bet would be to let it drive a audio transformer.  That way the chip can still add whatever common mode signal it wants to.  You can ground one end of the secondary and use the other end as a single-ended audio signal you can feed directly into a power amplifier.