The main difference between a mug and a coffee cup is that the coffee mug is normally taller and coffee cups tend to taper near the bottom. The first and most striking difference between the two is the distinctive shape of coffee mugs vs coffee cups. A coffee mug, by definition, is taller, and they tend to taper much less (if at all) towards the bottom. On the other hand, the bowl type coffee cup has a wider mouth and a narrower bottom.
Most types of coffee cups are also in general more fragile than a mug. Coffee cups tend to be made from porcelain which, although superb for flavour, heat retention and achieving the correct intake of air when drinking, is brittle and breaks easily.
Coffee mugs tend to be more robust: which is why you tend to see them less in cafes and much more in homes, offices, and workshops.