What is the difference between arithmetic and mathematics?

It is a useless exercise to establish difference between arithmetic and mathematics. Mathematics is abroad term that arithmetic as one of its branches.

To move further in this topic it’s important to define both these terms separately:
Mathematics

Mathematics can simply be defined as a method of studying the relationships between shapes, quantities and numbers. It use symbols, signs, theorems and formulas and includes calculus, trigonometry, algebra, arithmetic and geometry.
Arithmetic

Arithmetic is a specific branch of mathematics that mainly deals with the multiplication, division, subtraction and addition. It uses numbers for calculations.

One basic and most highlighted difference between arithmetic and mathematics is that mathematics is about theories but arithmetic is all about calculations and numbers. When different branches of mathematics club with other technical studies like applied physics, particle physics, descriptive statistics, inferential statistics etc then it becomes ‘Real Mathematics’ with concrete applications in real world, otherwise mathematics is just a study of variables and their relations.
 
 
 
 
 
 
 
Arithmetic and the Ghost Story

Many thinkers classify arithmetic as study of logical ghosts. To understand this let me share a story with all readers. In some rural areas of India, a father and a son were moving towards a city on their bike. On the way father started telling his son about the stories of ghosts that were very popular in their village. Suddenly son asked his father that do you believe in ghosts. The sudden reply of father was “Of course not”. But on their way a thought came into father’s mind that if he believe in numbers he have believe in ghosts also, because a ghost is nothing but a non-concrete entity that cannot be touched and that have no mass or weight. So, what is a number? It is simply a symbol but with an attached and defined meaning. So a number study is just like a ghost study, but here your ghosts are defined by notations. For example- Egyptian number system was alien to all of us until we haven’t cracked the meanings associated with each number in Egyptian number system.