I just ran another gravity test using the hydrometer and it's down to 1.003, so it's obviously finished. I had the understanding that the refractometer would read the gravity more accurately than the hydrometer, this is a major difference.
Not so. They are good for sugar but get all jacked when alcohol is present. I have also read that refractometers that are dual scale (brix and sg) are not accurate when it comes to the SG scale as it isn't a true 1:4 ratio. Cheaper refractometers are also not as reliable in my experience. I have one that will read different from sample to sample, but it sounds like yours is consistent!
Edit to add - Hydrometers are calibrated at a given temperature and require a calculation to get the true value. Every hydrometer I have used is calibrated at 60°, but I hear there are some that are calibrated at 68°. Either way, it should be printed on the scale.
To make your life even worse, hydrometers aren't always accurate and require testing in distilled/DI/RO water at the proper temp to see how far off it is if at all. In my opinion, this only matters if you are using multiple hydrometers. If a hydrometer is off, the difference between OG and FG will still be constant, just not as accurate.