Euclidean isometries in $\mathbb{R}^3$ are compositions of a translation and an orthogonal transformation. Each Euclidean isometry is a surface isometry that preserves length of rectifiable curves, but the converse is not true. A length-preserving isometry between $\mathbb{R}^3$ surfaces in $S_1$ and $S_2$ is a bijection that preserves dot products of tangent vectors.
For local isometries there is plenty of example, but I need an example of surface isometry that is not an Euclidean isometry.