Arithmetic is (according to the Oxford English Dictionary) 'the science of numbers; the art of computation by figures'. Arithmetic is important because it is the mathematics everyone starts with. Arithmetic teaches one how to count, add, subtract, multiply, divide, and how to do those things to use in the real world for many important and helpful things.
An important idea in arithmetic is The Fundamental Theorem of Arithmetic. It states every natural number greater than one is either prime or is a product of prime numbers. There are at least four ways to prove that, and every mathematician should try it at least once (and normally does so in college/university).
Number theory, or higher arithmetic (as opposed to basic arithmetic), is unsuprisingly about the study of numbers, at least the natural numbers and possibly integers, but not necessarily any more complicated types of numbers unless applied in other areas of mathematics. Number theory is often considered a graduate-level subject, but in ancient or Classical Greece (and even the early modern highly civilized world, when most college/university students and intellectuals read Euclid's Elements, which includes number theory) it was one of the main areas of study and is the most important mathematical foundation (of ideas) besides reason & logic.
Mathematicians should have good arithmetic skill, and pure mathematicians should study number theory, since pure maths' main foundation of ideas is number theory.