What's the difference between a 9v DC battery and a power supply?

The difference is that the battery has a higher impedance, especially at the frequencies you are using, than the nicely regulated power supply.

Somewhere in your circuit, probably in the early stages of the audio amplifier, it is making the assumption that ground and power are equivalent for AC signals. From this you can derive that the impedance of the power supply is assumed to be 0. Of course no supply has exactly 0 impedance, but the regulated power supply is a lot closer to this than the battery, and that difference is enough to upset your circuit.

To fix this and to verify this is really what is happening, put a decent size cap across the battery. Try something like a few 100 µF at least, but a mF or more would be better. It would also be a good idea to put a 1 µF ceramic cap across the power feed right where it enters the circuit. That will lower the impedance at high frequencies, something the large electrolytic cap can't do.

If a cap across the battery fixes this, and I think it will, then your circuit was poorly designed. Caps on the power supply should have been included as part of the circuit in the first place, so that it doesn't rely on the impedance of some external source.

A battery will output most of the time a constant voltage. A PSU converts 110/220 AC voltage into your 5VDC (or whatever voltage), and that process isn't perfect, so a PSU can't output trully constante voltage. Here is a image that ilustrate the AC/DC convertion process.

_{(source: uonbi.ac.ke)}

So when you are working with amplifier, your ripple voltage will also get amplified, and that may cause trouble on your circuit. A way to maybe work this problem out is to put a capacitor in your PSU output, the higher its value the best, but that may have some other issues, like taking some time for your voltage to get stable, cause it will need to charge the capacitor for that.

For more info on the topic I suggest getting a book that talks about voltage regulation and power supplies. Most basic eletronics book will have this.

And of course this dosen't happen to batteries cause they output "pure" DC voltage, not a transformed from AC one.