The only difference that I know about is that paste is usually in a paste form and applied with an applicator. The flux is generally in liquid form and may be applied with a different applicator. A lot of solder has the flux within the core of the solder. There are different types of flux, some are water soluable, meaning they will wash off with water. The oldest and the most common for small soldering jobs is not washable with water, but cleans well with alcohol. Fux for plumbing is different and should not be used on electronic or electrical connections and it can cause corrosion of the joint.