When I'm walking through a mall and they're playing Justin Bieber, that's noise. When I'm walking through a mall and Justin Bieber is actually there, that's a disturbance.
A disturbance is like a transient that decays arbitrarily close to zero in finite time.
Noise has an infinite time span.
In DSP terms an FIR filter will respond to a disturbance and the output will disappear in finite time. An IIR filter will have an output that does not decay to zero with a disturbance on the input.