Q:

How do Power and Strenght values correlate? 

When creating a non-light and giving it an emission shader, what values of "Strength" are equivalent to given values of "Power" in an area light? 
Not a duplicate of Understanding light power values.  This isn't talking about options in lights, but options in emission shaders that aren't lights on their own. 

A:

I did some testing.  You can use this as a reference sheet :). 

Power : Strength ratio is 10:2,5 (for 1m² area)

Area lights use Power (W), Mesh lights use Strength
Mesh light used - default plane with Emission shader

Intensity and size relationship

Area lights - maintain their intensity with increased size. 
Mesh lights - increase their intensity with increased size

Power:Strength reference sheet 

I tested these as 4096x4096 images and still couldn't tell the difference. 
Sorry for the color banding and compression artifacts. 
Noise is a tiny bit higher for Mesh lights.