What's the difference between atomic orbitals and energy levels?
Neon's atomic number. But then what are the atomic orbitals, like s,
  p, d, and f. Where do they come in? Are they completely different from
  the energy level?


Atomic theory was developed in stages almost a century ago. Bohr's model is the simplest version of hydrogen atom. General chemistry textbooks write rarely mention the Sommerfeld's model from which the concept of orbital angular momentum originated on elliptical paths (recall that a circle of Bohr's model is a special ellipse). Sommerfeld model model explained more features of the hydrogen spectrum. After Sommerfeld the next person was Schrodinger who proposed the H atom in terms of probabilities of finding electrons rather than fixed orbits. 

In short, s, p, d, f come into play in more advanced models of Sommerfeld. Bohr was the first one to talk about electronic configurations for multielectron atoms.

Sommerfeld' model