There is a classical example with $NW(f)\backslash \overline{R(f)}\neq\emptyset$, the so called Bowen's eye-like attractor (see the paper by Baladi, Bonatti and Bernard: Abnormal Escape Rates from Nonuniformly Hyperbolic Sets):

![alt text][1]

Every point on the dark curve is nonwandering, but only the two corners are recurrent (in fact these two are fixed).

Why is the theorem called Fundamental? One reason is that it is the 'correct' setting of $C^1$ stability conjecture (see the discuss [here][2]).

Theorem: Let $f$ be a diffeomorphism on a closed manifold $M$. Then the following are equivalent: <br>
1. the map $f$ is  structurally stable;<br>
2. $\mathcal{R}(f)$ is hyperbolic;<br>
3. $f$ is $\mathcal{R}$-stable.<br>

This modern version is very succinctly comparing to the version using the nonwandering set, which involves with no-cycle condition and transversality condition.

Also I copied a few words from the paper mentioned in Barry's comment:

'The theorem is fundamental in the sense that it deals with the basic question of the field. It is also fundamental in that it encompasses such big ideas in such a small, concise statement.' 

Compared to *Fundamental Theorem of Arithmetic* and *Fundamental Theorem of Algebra*, Norton wrote: 

'the space on which the dynamics take place, can be decomposed uniquely into its basic dynamical parts: points whose dynamics can be described as exhibiting a particular type of recurrence, and points which proceed in a gradient-like fashion.'


  [1]: https://pfzhang.files.wordpress.com/2012/12/bowen.png
  [2]: http://www.scholarpedia.org/article/Talk%3ASmale_horseshoe