Q:

Why do we call primes, and not the number one, *the atoms of numbers*?

The fundamental theorem of arithmetic asserts that we can produce every composite number from a unique set of prime multiplicands, so long as none of those primes equals one. 
Consequently, some mathematicians call the primes the atoms of numbers. However, we can define every number, prime and composite, by adding a unique number of ones; so the analogy between atom and the number one seems more robust than the analogy between atom and prime number does.
Why do mathematicians call the prime numbers, but not the number one, the atoms of numbers?

A:

Adding atoms is just increasing number of moles,while multiplying them makes molecules.

A:

With just one kind of atom, the world wouldn't be so grand and beautiful, would it?
Mathematicians prefer to talk about primes for the same reason that they prefer to do number theory, not with real numbers ($\Bbb R$), but with only integers ($\Bbb Z$): Only in the latter case is there something interesting to discover. 
The sum of ones does not provide any interesting information about a given number $n$. While it's true that $$n = \underbrace{1+1+\dots+1}_{n}$$ we still need to know the number of ones in order to make the sum become $n$. In other words, we need $n$ in order to "construct" $n$. In any event, this is just a trivial sum. In contrast, when multiplying primes, the prime numbers themselves, when multiplied, "construct" the number $n$. To get 10, the prime factorization $2\cdot5$ contains all the information. 
And this is not a trivial construction. On the contrary, this structure of primes is very interesting to observe and study, just like the atomic structure of a real-world object is interesting to observe and study. There is so much to know (and still yet to be known) about primes and prime factorization, and the concept is so fundamental to more abstract algebra.
(I also very much liked the analogy of molecules from kingW3.)

