Batz for taking readings after the fermention has starterd you have to take into account there is alcohol in it, the refraction of light will not be a true reading for the amount of sugar in solution.
But if your either a mathematical proffesor or have brewing software you can get the correct reading.
Go to the beersmith refractometer tool, the first thing you'll set is the calculation to finished beer, then put in your refractometer reading for the finished beer.
Now what you do is adjust the one that says hydrometer reading untill the original gravity reading is what your original gravity was.
the hydrometer reading under the brix reading is then your final gravity.
Iam not sure if that makes sense so go grap a beer an try it. If you now the beers OG you can take a brix reading and get the final gravity, but you'll just need software to do some calculations which are almost impossible for he human brain.
Anyway simple answer to why FG would be diff is because there is grog in the sample.
Jayse