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.

There's a BIG difference between soldering paste and solder paste. Soldering paste is a flux. Flux is available in either liquid or paste form and used as the cleaning agent to remove contaminants while soldering.

Solder paste is a mixture of tiny balls of solder and flux, used for mounting SMT devices in place of regular solder. It's usually applied rubber-stamp fashion (e.g., silk screen) and a board is populated with parts and run through an oven to melt the paste on all the parts simultaneously. It's the SMT version of wave soldering. Solder paste is expensive stuff and is available in small tubes for hobbyist use or for repair.

Dean