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

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

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.)

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.