The difference between dye and paint is the difference between dye and pigment because paint is colored with pigments. Dye consists of organic small molecules in aqueous or alcoholic solution. Fabric is dipped into the solution and absorbs the dye. As the solvent evaporates, the dye molecules stay bound to the fabric. Many dyes consist of what are called azo compounds. These are organic compounds with a nitrogen nitrogen double bond. So usually there will be two or more aromatic rings joined together by azo groups. These types of compounds usually have very strong light absorbing properties, so they are often brightly colored. Some examples of dyes are methyl red and bromocresol green.
Paint is a more complex mixture. The colorant in paint is pigment. Natural pigments include various minerals and salts but synthetic pigments are also made. Another component of the mixture is the binder. This is what causes the paint to dry and adhere as a thin film and it is usually composed of some type of polymer resin. Finally, all of this is dissolved in a solvent, usually water or an organic solvent. Some examples of pigments are chromium (III) oxide (Cr2O3: green) and cobalt (II) stannate (blue).
The main difference between paint and dye is that the dye actually dissolves in the solvent to ultimately adhere to the fabric. The pigment in paint is not soluble in solvent. So paint is really a suspension of very finely ground pigment particles in the paint mixture. The pigment particles adhere to the binding agent which in turn adheres to the wall or painted surface. The other major difference between the two is that dyes are organic molecules while pigments are more salt and mineral (ion) based.
I cannot draw any dye structures here but see the link below for more information.