These two methods are used to measure the moisture content in samples.
Loss on drying is a simple procedure where the sample is heated to a specified temperature above the boiling point of water (usually 105 C). From the mass before and after heating, we can calculate the loss after heating. We assume that all the mass lost is water, but we have no way to verify that fact from the test data. A "moisture balance" is an analytical balance that is designed to make this measurement automatically.
The Karl Fischer method uses a chemical reaction to measure the water content. The amount of water is calculated by measuring the amount of reagent required to react with the water. There are automated instruments that perform this test.
For many samples, the two methods produce identical results. However, the sample contains some volatile material (solvents), then the loss on drying value will be larger.