In residential power system jargon, a "separate circuit" means that you have a separate set of wires, with its own circuit breaker, running from the circuit breaker panel to the outlet.
This prevents other electrical devices on that same breaker from pulling down the voltage under heavy load - for example when an air conditioner compressor starts up.
Probably more importantly, it eliminates the risk that a large amplifier, which may draw a huge amount of startup current, may cause a momentary overload which may affect other equipment.
It may also slightly reduce the effect that especially powerful noise spikes generated by equipment on other circuits will have on devices connected to that circuit.
HOWEVER, in most homes, all of the circuits in your home are connected to the same power company transformer, via the same drop cable to the house.
So, in fact, they aren't really "separate circuits", but merely separate branches of the same supply feed... and they aren't much isolated from each other from the perspective of line noise.
That said, I rarely encounter audio gear that seems to be seriously affected by line noise... and, when I do, the result is always relatively obvious.
I use a very reliable non-sacrificial series mode surge suppressor on my audio gear for protection against line surges and lightning strikes - which also happens to provide a significant degree of filtering against high frequency line noise...
But, to be honest, I can't claim to hear a difference when it's disconnected... (because I don't have a problem with line noise to begin with).
In most cases, when line noise is a problem, you can actually hear it... for example, you hear a "pop" from your speakers when your A/C turns on, or a faint "zzzz" when a certain lamp is lit.
And, in those situations, a noise filter, or even an isolation transformer, can provide a very effective solution.
HOWEVER, you should beware of claims that various products will provide subtle or marginally audible improvements by reducing noise that wasn't audible to begin with.
While there is sometimes truth to that claim, it is often simply wishful thinking, or the power of suggestion.
I would be remiss if I were to point out that a high quality commercial isolation transformer of sufficient current rating isn't going to do any harm...
But some of the many audiophile gadgets that make this sort of claim can actually make the noise situation worse...
And some, while they won't affect the performance of your audio gear at all, can be a serious threat to the condition of your wallet...
(This is really the sort of thing that screams out for a double-blind test to confirm what we may think we hear.)
