I knew that the debate would start :) This is probably the most contentious question in helicopter operations. it's right up there with the "equal transit time" theory of lift generation a few years back. The only agency I am aware of that defines them as different is Transport Canada. Instructors, pilots, WikiPedia, training videos, books and blogs all get this wrong often. It seems to be very common in the US for people to refer to vortex ring state as "settling with power".
Tom, I know you said you already know what vortex ring state but I think that the "why are they different" explanation is better when both are contrasted together. Please excuse me for repetition.
They are different.
Vortex ring state (VRS).
VRS ring state is a purely aerodynamic condition. In normal flight, whether climbing, descending or cruising, the relative airflow striking the blades is principally from ahead. In a developed VRS, the helicopter descends into it's own vortex which is "recirculated" as the helicopter descends faster than the vortex descends and results in the recirculated air being drawn up, then almost vertically down into the rotor.
This causes a critical increase in the angle of attack and a corresponding loss of lift causing the helicopter to descend faster and faster up to rates of 6000 feet per minute.
In order for the helicopter to descend into its own vortex, it must meet three conditions at the same time (typical values for most helicopters):
less than 30 knots airspeed;
greater than 300 feet per minute descent;
power applied to the rotor.
Removing anyone of these gets you out of VRS but there's a catch. If you try to reduce the rate of descent by increasing collective and therefore power, which would be the instinctive reaction of someone not trained to recognise and get out of the state, you increase the strength of the vortex and the recirculation and make the problem worse.
The only way to stop the rate of descent in VRS is to use the ground, which you really don't want to do at up to 6000 feet per minute and will probably spoil your plans for the weekend. However, you DO have power. You can use all of it until you hit the limits and still you won't stop that descent.
To recover, remove power by entering auto-rotation or increase airspeed to fly out of the vortex. Auto-rotating out of VRS can take a long time and use a lot of height in the recovery so the accepted and taught technique is to fly out of it, any direction will do, which takes a second or two to recover.
Settling With Power (SWP)
SWP is not an aerodynamic effect.
At it's simplest, it occurs when you need x amount of power to stop a descent but, for a lot of variable reasons, you have less than x power available. Increasing power results either in exceeding the engine or power train limits or stalling the rotor as drag increase exponentially and exceeds the power available to drive the rotor.
The slower a helicopter flies, the more power it requires:
From the AOPA blog.
When you get into the low speed regime, you can see that the power required rises rapidly.
Imagine you are coming into a hover. You are travelling slowly at about 20 knots but have a low rate of descent, not enough to enter VRS. If you keep going forward, you will be OK. But, our imaginary pilot makes a mistake and turns into a 15 knot tailwind. The airspeed drops to about 5 knots so the lift produced drops by the square of the speed change and the power required rises rapidly.
The helicopter will very quickly develop a high rate of descent. The pilot increases collective to compensate but finds that the engine cannot produce enough power to maintain lift and power the tail rotor which is also working harder since you increase the torque by increasing power. It is possible to simply run out of power. If the pilot is lucky, they will be unconscious and found with the collective lever buried in their armpit.
SWP nearly always happens close to the ground and once in it, there is no recovery (insert suitably large number like 99% of the time). You have no spare power to reduce the rate of descent or to increase airspeed to increase lift and fly away.
SWP is normally a result of pilot error (as above), mechanical problem leading to unexpected reduction in power available, wind shear causing unexpected reduction in airspeed or other environmental effects reducing lift available (coming into a hot and high hover for example).
Perhaps the easiest distinction to make is that you are already using all of your available power when you enter SWP. There is nothing left to give!