I confess to being confused by Natasha's answer, and can only presume that these words have a different meaning where she lives. Natasha's description of 'cup' sounds more like a glass, whereas her description of 'mug' covers both 'cup' and 'mug'.
This is what cups and mugs have in common:
Both are usually ceramic
Both have handles
Both are generally used for hot drinks - tea, coffee, chocolate
(The exception to this is a baby cup, which is generally plastic and used for cold drinks - simply because it's more practical to give a baby or toddler a drink in a vessel that isn't glass but does have a handle).
So if both mugs and cups are usually ceramic, used for hot drinks and have handles, what is the difference? Simple - it's one of size and shape:
Cups come in a variety of shapes of sizes. An espresso cup is very small, whereas a cappuccino cup is wide and open, almost like a bowl. A tradition teacup has a dainty narrow base and then widens out towards the top.
Mugs are always cylindrical in shape and are generally larger than a cup. They're less refined, but being cylindrical, they hold more liquid.
'Mug' is also a more specific term. You could refer to a mug as a 'cup', but a wide cappuccino cup or dainty espresso cup are both 'cups' - definitely not mugs.