If you're talking about the actual objects, there is a difference in shape. A plate is almost flat, while a dish is deeper and more concave. For example, you'd serve a sandwich ON a plate, but you'd serve soup IN a dish. (Note the difference in prepositions, indicating that a plate is a surface while a dish is a container).
With a slightly different meaning, 'dish' is also used for a variety of shallow cooking and serving receptacles. For example, you can get ovenproof dishes of various shapes, often oval or rectangular and a few centimetres deep. You might cook lasagne in the oven in a dish of this type and then serve it on the table from the same dish.
The other difference, of course, is that 'dish' can refer to the food itself e.g. "Bulgogi is a popular Korean dish made with beef and soy sauce". 'Plate' cannot be used in this way.
I hope that helps.