Simply multiplying ABW by 1.25 to get ABV is an approximation (and not a very accurate one)
Let's assume that you have 1 liter of beer with a gravity of 1.010 at 20C, and that the beer has 5% ABW.
At 20C, that beer will have a weight of 1010 grams (by definition).
5% of 1010 grams is 50.5 grams, so the beer will contain 50.5g of ethanol.
Ethanol has a density of 0.78945 Kg / Liter at 20C so the volume of ethanol would be 0.064 liters (50.5 / 0.78945 / 1000)
This translates to 6.4% ABV rather than the 6.25% estimated by the formula ABW * 1.25 = ABV.
Now assume that you have 1 liter of beer with a gravity of 1.020 at 20C, and that the beer has 5% ABW.
At 20C, that beer will have a weight of 1020 grams (by definition).
5% of 1020 grams is 51 grams, so the beer will contain 51g of ethanol.
Ethanol has a density of 0.78945 Kg / Liter at 20C so the volume of ethanol would be 0.065 liters (50.5 / 0.78945 / 1000)
This translates to 6.5% ABV which deviates even more from the formula, and illustrates that the gravity of the wort must be taken into consideration when accurately converting ABW to ABV
-a.