$ U(1) $ and $ SO(2) $ are locally equivalent.
Both $U(1)$ and $SO(2)$ are geometrically the circle $S^1$.
$U(1)$ is the set of complex numbers (or $1\times 1$ complex matrices) such that $z\bar{z} = \bar{z}z = 1$. That is, the set $e^{i\theta}$ for real $\theta$.
$SO(2)$ is the set of rotations in $\mathbb R ^2$. The rotations $r_\theta$ form a group under composition.
These two groups are isomorphic, and the isomorphism is just the one described by Jyrki: for a complex number $z$ in $U(1)$ the corresponding element in $SO(2)$ is the rotation by the angle which is the argument of $z$.