Is mg/m^3 same as ppmw (CO2)? I'm trying to convert $\ce{CO2}$ ppm (volume) in the air into ppm (weight) in the water. 

I found this formula


  $\mathrm{mg \ m^{-3}}$ = (ppmv)(MW) / [($0.08205$) (¦$\mathrm{K}$)]


where,

ppmv = air pollutant concentration, in parts per million by volume 

$\mathrm{mg \ m^{-3}}$ = milligrams of pollutant per cubic meter of air 

$\mathrm{K}$ = atmospheric temperature in degrees Kelvin = $273.15$ + $^\circ C$

$0.08205$ = universal gas law constant in $\mathrm{L\ atm \ mol^{-1} K^{-1}}$ 

MW = molecular weight of the air pollutant (dimensionless) 

$\mathrm{atm}$ = absolute atmosperic pressure in atmospheres 

$\mathrm{g \ mol}$ = gram mole

I've read the $\mathrm{mg \ m^{-3}}$ is a subunit of ppmw, so is it correct to say $\mathrm{mg \ m^{-3}}$ is ppmw?
No. ppmw means parts per million by weight (mass). This is not the same as mg/m3, which is milligrams (mass) per cubic meter (volume).

For example, in a gas of very low density (e.g. $\ce{CO2}$ at 10-6 Pa), 1 mg/m3 could exceed the weight of $\ce{CO2}$, i.e. would be greater than 900,000 ppmw.

One measure has to do with mass in a given volume, the other measure is mass in a given mass of another substance.