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: 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.)
A: You are right. But you must realize that the way $1$ generates the numbers by addition is easier to study than the way the primes generate the numbers by multiplication. 
The study of the first you finish by elementary school. The other not so fast. 
As you said, the generation of the numbers by the one by addition is more robust. That is why it is often used as a way to define the numbers. Almost never you find a definition of the numbers beginning form the primes. 
A: Adding atoms is just increasing number of moles,while multiplying them makes molecules.
A: The word "atom" is not ideal for prime numbers.
Atom literally means "indivisible", so in that sense, the number 1 is as hard to divide into factors as $2,3,5\dots$ (and incidentally, the ancient Greeks regarded 1 as prime number).
However, the modern view prioritizes the algebraic structure, so the better word would be "generator" instead of atoms, and here, you can tell that 1 does not contribute anything to a product.
A: Prime number is the number that has only two factors, $1$ and itself. 
The number $1$ has only one factor for that it is not a prime number. 
A: Your question actually includes its own answer in one sense; you wrote:

...we can define every number, prime and composite, by adding a unique number of ones....

In other words, you can define a number by that number (of ones.)  This sort of self-reference is hardly a definition of a number in any sort of rigorous sense.  Not to mention that this approach entirely fails to define the number "one."  ("A one is one one."  Or, "$2=1+1$, and $1=1$.")
Pointing out a specific number of apples or blocks, as you would teach a child, is, of course, a perfectly valid ostensive definition, but prime numbers are a different sort of construction entirely.
And, as naslundx points out, it is a much more interesting way to look at numbers.
