Both paint and dye create colour on leather, but the one you choose depends on the kind of leather, and how you will use it. Is the leather a pair of shoes you wear frequently, or a piece of art you will display with care?
Paint sits on the surface of leather, while dye penetrates leather. Leather paints are chiefly acrylic, while dyes are either spirit (alcohol) or oil based.
Even the most flexible paint is subject to lifting, cracking, and flaking over time. So, dye is better suited for items of clothing, wallets, belts, and other high-use items.
Paint is more forgiving on finer work. You may apply it with small brushes, even fine-point paint markers.
Because paint covers the leather completely, you can achieve any colour you wish, including paper white and buttercup yellow. Dyes are far less vivid in colour, as the background colour of the leather comes through.
Professional leather workers use dye far more frequently than paint.
You can use dye to achieve full coverage--for example, dyeing a pair of shoes a deep black. You may also use dye to achieve authentic western, Spanish, antique, and Elizabethan effects, since dye was used in all those times and places.