An AC power supply rectifies DC power from AC, so there is necessarily a filter. You'll have rectifier switcing artifacts and low frequency AC noise to deal with. A battery does not have these issues.A battery has higher internal source impedance and has to be recharged or replaced, so those are the downsides to a battery. But it is pretty noise free and can be an attractive option, especially in a circuit like this that needs to be noise free and that has very low curent draw.