I have tried to use a refractometer when brewing. I can never get an accurate pre-boil specific gravity reading that consistently aligns with my post boil, cooled wort hydrometer reading. I spoke with another homebrewer who has the same issue. He ultimately decided to defer only to the hydrometer. Am I doing something wrong with the refractometer or is best to get a pre-boil sample, put in the refrigerator for awhile and then take a hydrometer reading? There would still be time to make adjustments by boiling off or adding DME if necessary.
The first thing to note is that your refractometer and your hydrometer are measuring different physical attributes of your wort. You need to account for these differences, and additional errors.
The refractometer is primarily measuring sugar content, through the difference in refraction relative to pure water. This is where calibration comes into play. Most instructions recommend calibrating against distilled water, and adjusting the refrac such that it measures 0 Brix|Plato. I've mulled the merits of calibrating against the brew water, to account for any content that might give me a false positive/negative, but that's for another time.
Because you're using a small volume on your refrac, the temperature is quickly brought to ambient, so this shouldn't be a concern. However, if you haven't mixed your wort well prior to taking a sample, you will not get a representative value - remember your first runnings are densest and will have a higher value than your last runnings at the unstirred surface. This and proper calibration should mitigate any errors you'll get with using a refrac.
Hydrometers measure the density of the solution, as calibrated against the density of water (usually at 15 or 20 C - this should be listed on the device or manual). This means that hydrometers are influenced by all things in solution, sugars or otherwise. Solids generally don't contribute as they precipitate out. However, you're measuring more than just sugar. Of note, "This method of analysis is most commonly a strewn with errors as it relies on the viscosity of a substance which is heavily altered by the temperature of the sample. As the sample is heated it will decrease in viscosity and as it is cooled it will increase in viscosity. For this reason it is very important to always note what temperature your sample is upon reading with a hydrometer." (Brew 1204, Lab Manual, 2015)
A rough conversion from SG (measured by hydrometer) and Plato (measured by refrac) is (((SG-1)/4)*1000). While not wholly accurate, it will likely be within an acceptable margin, given the errors that occur elsewhere in the process. This of course assumes the values were measured at the same temperature.
This is probably the biggest source of discrepancy, the temperature differential. We can assume that your refrac is essentially measured at ambient, (20-25 C?) but your hydrometer is at your wort temperature. It is therefore important to measure your wort temperature as you're taking your hydrometer reading. You can then use a calculator to determine what it would read at ambient. Alternatively, you can cool your sample to ambient and then measure your SG.
Pre-Boil vs Post-Boil
Your pre-boil gravity will inherently be lower than your post-boil gravity, as you are concentrating the wort over the course of the boil. This change in volume (your evaporation rate) will be moderately variable brew-to-brew, and generally ranges from 5-15%.
In order to predict values, we need to know at least three of the four variables: pre-boil volume, pre-boil SG, post-boil volume, post-boil SG. The fourth variable can be calculated using (C1*V1=C2*V2) where C is your SG (just use the post decimal digits, i.e. 1.060 = 60) and V is your volume (any units will do so long as you are consistent).
Summary / Biggest Source of Discrepancy
Here again, accurate measurement using a refrac and hydromter require thorough mixing of the wort to ensure you're getting a representative sample. This and accounting for temperature differences are the biggest sources of discrepancy between refrac and hydrometer readings. Since they don't measure the same physical parameters, you can still expect a small difference between them.
First, refractometers vary, their quality is different from model to model. Temperature range they can work properly is pretty wide, but it's there. Make sure you are not outside it.
Second, with your refractometer you should get manual and calibrating solution. Be sure to use these, without proper calibration readings will be off.
Third, of course pre-boil gravity will be lower than post-boil (unless you add water during boil). You can calculate expected gravity knowing volumes, of course. If your calculations are inconsistent with your readings, maybe it's the volume measure what's off, not refractometer? Or maybe you made simple mistake in calculation? Happened to me, once...