There are a number of differences between a cup and a mug, the most notable of which is that a cup is typically used for tea, while a mug is often used for coffee. Also, a mug is usually larger than a cup.
The Free Dictionary defines a cup as a small open container with a flat bottom and handle, while a mug is defined as a heavy, cylindrical drinking cup with a handle. It can therefore be inferred that while a mug is a type of cup, a cup is not always a mug. A mug can be more robust than a tea cup with a saucer.