I happened to be thinking about this today: how individual primes are related to prime distribution. For example, there are many theorems which seem to relate to individual primes rather than the aggregate of all primes taken together, such theorems as Fermat's Little Theorem, the fact that a prime is of the form $3n+1$ iff it can be written as $x^2+3y^2$ (equivalently $x^2+xy+y^2$) for coprime integers $x,y$, or the Quadratic Reciprocity Law.
One thing I have managed to spot is the fact that theorems in the former category seem to come from algebraic number theory whereas theorems about the distribution of primes come from analytic number theory. This would make sense intuitively since algebra is kind of like a discrete version of analysis. Anyway, I don't think I can get far using this kind of (intuitive) thinking alone and was hoping that someone more experienced than I (after all, I have not studied maths at university yet) could shed some light on this?