What is the difference between these measurements and is one better than the other for any particular use?
Which measurement scale do you use, why?
I use both because my refractometer reads Brix (plato), but I "grew up" on SG and I often take my FG with a hydrometer, so I'm used to SG. There is also a good rule of thumb for determining alcohol content with SG. Drop the 1 and move the decimal place to the second to last place and that will approximate ABV. In other words, if you brew a 1.067 beer and get average attenuation, you will have approximately a 6.7% ABV beer. To do the same thing with brix you multiply by .4, but that is less accurate and harder to do in your head.
The technical difference between the two is that Brix and Plato are a measure of the percentage of the solution that is sugar. So 11 degrees brix is a solution of 11% sugar compared to the weight of the water. Specific gravity is a measure of the density of the solution divided by the density of water. See here for a more detailed explanation.
Specific gravity AND degrees Plato or degrees Brix is a measurement of density; based upon the amount of dissolved solids in a solvent. In the case of wort we are talking about sugars in water. Sugar being the main component, but other solids have dissolved in the water too, effectively increasing density, but to a much lesser extent than the sugars contribution. This is important because it does relate to total attenuation.
Hydrometers often have a multiple scales on them where you can read SG OR Plato on the same hydrometer.
Refractometers tend to be mostly in Brix/Plato scales.
Neither one is really any better than the other. Its just a convention of preference depending on the equipment you use to measure the density of your wort/beer.