Are $S^1$ and $\mathbb{R}/{\sim}$ the "same thing"?

The relation is not given by that partition. As said in the comments, the relation is:

$x \sim y \iff x=y \mod 2\pi$

Now, consider the following map:

$f: \mathbb{R} \rightarrow S^1$; $x \mapsto e^{i x}$

Since it takes equivalents to the same image, the induced map:

$\tilde{f}: \mathbb{R} /{\sim} \rightarrow S^1$

is continuous.

Now, take the map $g: S^1 \rightarrow \mathbb{R}/{\sim}$; $e^{ix} \mapsto [x]$. It is obviously well defined, and easily seen to be continuous. Note that $g$ is the inverse of $f$. Therefore, the spaces are homeomorphic.