An irational rotation $f$ is arbitrarily distorted in $\text{Diff}^{\infty}S^2$, this was proved by Calegari and Freedman http://arxiv.org/pdf/math/0509701.pdf and in a more general setting by Militon, see http://arxiv.org/abs/1005.1765. This means that for any diverging sequence $g(n)$(diverging as slow as one wants), there is a subsequence $f^{n_i}$ and a finite set $S$ in $Diff(S^2)$ generating a group $G = <S>$ such that $\{f^{n_i}\} \in G$ for all $i$ and that $l_S(f^{n_i}) \leq g(n_i)$. ($l_S(g)$ is the word length of $g$ in the generating set $S$). To make it a little bit more clear, let's consider the following example: Let's take the translation $a: x \to x+1$ in $Diff(R)$. If we consider the subgroup $a:x \to x+1$, and $b : x \to 2x$ in $Diff(\mathbb{R})$, observe that $b^{n}ab^{-n} = a^{2^n}$, this implies that $a$ is at least exponentially distorted in $Diff(\mathbb{R})$, because we can write $a^{2^n}$ as a product of $O(n)$ elements in $a$ and $b$.

One condition that ensures that a given $f: S^2 \to S^2$ is conjugate to a rotation is that the sequence of derivatives of powers of $f$, $(Df^k)_{k \in \mathbb{z}}$, is a bounded sequence. The idea is that in that case, one can take an arbitrary riemannian metric $g$, take and averaging sequence $g_n = \frac{1}{n}\sum_{i=1}^n ({f}^n)^*g$, and observe that the bounds in derivatives imply that $g_n$ has a subsequence converging to a metric $g_{av}$ invariant by $f$. If $g_{av}$ is smooth, by the uniformization theorem, $g$ is conformally equivalent to the standard $S^2$, where $f$ is acting by rotations.

Using these two ideas, one can get a group theoretical way of characterizing any $f: S^2 \to S^2$ that is conjugate to a rotation: $f$ is conjugate to a rotation if and only if for any subsequence $(f^{n_i})_i$ of powers of $f$ there exist a subsequence $m_k$ of $n_i$ and a finite set $S$ of diffeomorphisms generating a group $G = <S>$ such that $\{f^{m_k}\} \subset G$ and $l_S(f^{m_k}) \leq k$. For more details, see the introduction of my paper http://arxiv.org/abs/1307.4447.

As rpotrie mentioned, it might be of interest to you some diffeomorphisms called "pseudorotations", that can be defined as diffeomorphisms of $S^2$ having exactly two periodic points. Birkhoff conjectured that any real analytic measure preserving pseudorotations is an actual rotation. There are many counterexamples in the $C^{\infty}$ category, in the case that the rotation angle is not diophantine, all of them as far as I know constructed by the same method. See http://www.math.psu.edu/katok_a/pub/Herman-survey-horo-corrected.pdf for a survey on this kind of constructions. There is a very positive result known as Herman's last geometric theorem in the case that $\alpha$ is diophantine. See http://www.math.jussieu.fr/~bassam/herman_ann_ens_second_revision.pdf. Using this method one can also construct diffeomoprhisms of $S^2$ that are arbitrarily distorted but not conjugate to rotations.