The operator-norm distance between two rotations is a chord length

Idea. Rotations are multiplications by $e^{i\alpha}$ in $\mathbb{C}$, so the distance between two rotations is the distance between two points of the unit circle.

Step 1: reduce to $R_\gamma-I$. Since $R_\alpha=R_\beta R_{\alpha-\beta}$ (composition of rotations adds angles),
$$R_\alpha-R_\beta=R_\beta R_{\alpha-\beta}-R_\beta=R_\beta\,(R_{\alpha-\beta}-I).$$
For an orthogonal $U$ and any $B$ we have $\|UB\|=\|B\|$ because $|UBx|=|Bx|$ for all $x$. Hence $\|R_\alpha-R_\beta\|=\|R_\gamma-I\|$ with $\gamma=\alpha-\beta$.

Step 2: compute $\|R_\gamma-I\|$. Identify $\mathbb{R}^2$ with $\mathbb{C}$, so that $R_\gamma x=e^{i\gamma}x$ and $(R_\gamma-I)x=(e^{i\gamma}-1)x$. Then $|(R_\gamma-I)x|=|e^{i\gamma}-1|\,|x|$ for every $x$: the matrix scales all vectors by the same factor, so its operator norm equals this factor.

Step 3: the modulus. Using $e^{i\gamma}-1=e^{i\gamma/2}\bigl(e^{i\gamma/2}-e^{-i\gamma/2}\bigr)=e^{i\gamma/2}\cdot2i\sin\frac\gamma2$,
$$|e^{i\gamma}-1|=2\left|\sin\frac\gamma2\right|.$$
(Alternatively, $|e^{i\gamma}-1|^2=(\cos\gamma-1)^2+\sin^2\gamma=2-2\cos\gamma=4\sin^2\frac\gamma2$.) This proves the formula.

Consequences. The maximum of $2|\sin\frac\gamma2|$ is $2$ (for $\gamma=\pi$): the diameter of $SO(2)$ in the operator norm is $2$, attained by $I$ and $-I=R_\pi$. In the Frobenius norm, $R_\gamma-I=\begin{pmatrix}\cos\gamma-1&-\sin\gamma\\\sin\gamma&\cos\gamma-1\end{pmatrix}$ has $\|\cdot\|_F^2=2\bigl((\cos\gamma-1)^2+\sin^2\gamma\bigr)=2\cdot4\sin^2\frac\gamma2$, so $\|R_\alpha-R_\beta\|_F=2\sqrt2\left|\sin\frac{\alpha-\beta}2\right|$, i.e. $\sqrt2$ times the operator-norm distance.

Check. $\alpha=\frac\pi2$, $\beta=0$: $\|R_{\pi/2}-I\|=2\sin\frac\pi4=\sqrt2$. Indeed $R_{\pi/2}-I=\begin{pmatrix}-1&-1\\1&-1\end{pmatrix}$ maps $e_1\mapsto(-1,1)$, of length $\sqrt2$.

Remark. The quantity $2|\sin\frac{\alpha-\beta}2|$ is the length of the chord of the unit circle subtending the angle $\alpha-\beta$; it is at most the arc length $|\alpha-\beta|$. So the map $\alpha\mapsto R_\alpha$ is $1$-Lipschitz from $\mathbb{R}$ (with the usual distance) to $(SO(2),\|\cdot\|)$.

Lesson. Use the complex-number model for $2\times2$ rotations; norms of "complex multiplications" are moduli.