There is no right answer to this question. Solid wood, especially the quality hardwoods are beautiful in their own right, but you can only get certain exotic wood patterns in veneer. Modern machine cut veneers are fairly thin between 1/50th and 1/32nd of an inch. Veneers on antique furniture made before the second industrial revolution (1850) tend to be thicker up to 1/4 inch because they were hand cut.
Many people think veneers evolved as a means to create furniture for less money, when in reality it was quite the opposite. Veneering was and still is a method of furniture construction, that allows for very elaborate decoration. Thin layers of bone, shell, and precious metals as well as wood have all been used to create highly priced and highly prized pieces of work. In many cases, veneers were the only possible way to use certain kinds of wood. Burls, for instance, are too weak to use as furniture components unless they are applied over a stronger wood as an undersupport. Some species of wood are so rare that you would only find them as a veneer.