#1 diesel is basically kerosene. K-1 Kerosene, #1 diesel, and jet fuel (JP4) are very closely related to each other, so closely that they can be interchangeable in most situations. #1 and JP4 have higher allowable sulfur than K-1. #1 diesel and K-1 have lower energy content (about 5%) than #2 diesel (and JP4) and is slightly more expensive. However, #1 diesel and kerosene does not gel like #2 diesel because the paraffin (wax) has been removed from the fuel. While this keeps the fuel flowing in cold weather, paraffin is an important lubricating compound found in #2 diesel.
Premium diesel is #2 diesel with a higher cetane number (usually a minimum of 51), increased lubricity, detergents that will clean injectors, and will also control and expel moisture from the fuel (demulsifiers), as well as corrosion inhibitors to prevent corrosion in the fuel pump.
I use premium diesel when I can find it, especially BP, Shell and Chevron branded diesel. Otherwise I'll use Power Service to increase the cetane, and Howes Meaner Cleaner to keep the injectors clean.