A painting is a type of artwork.
Other types are sculpture (any medium), prints (etchings, lithographs, woodblock prints, etc.), drawings, pottery, artistic embroidery or weaving -- anything made to be artistic and not just utilitarian.
In other words, "artwork" is a big category, and painting is just one sub-category within it. A painting, by the way, can be done in any one of a variety of paint media: oils, watercolors, acrylics, encaustics, tempera. A drawing could be done in charcoal, pencil, pastel, silverpoint, etc.
