The map (which actually goes from $THH^{C_p}$ to $THH^{hC_p}$) usually not an equivalence in the $p$-complete setting, at least if your input is genuinely a ring. You can detect the difference using a mapping cone; the mapping cone of this map is equivalent to the mapping cone of a map from $THH$ to the so-called Tate construction $THH^{tC_p}$ on it. For a ring $R$, the former is usually concentrated in nonnegative degrees while the latter rarely is (using homological grading).

However, there are some useful cases where it is an equivalence in sufficiently high degrees. These include, for example, the cases of $\Bbb Z/p$ and $\Bbb Z_p$, and there is a theorem of Tsalidis that (under some relatively mild extra assumptions) if this holds, then it also holds for the maps $THH^{C_{p^k}} \to THH^{hC_{p^k}}$. (This result played a critical role in Hesselholt-Madsen's calculation of the K-theory of local fields.)

If you were working rationally instead of in a $p$-complete setting, the homotopy groups of $THH^{hS^1}(R)$ correspond to "periodic" cyclic homology. In the $p$-complete setting instead, the homotopy groups of $THH^{hS^1}(R)$ are groups that are called $TF(R)$ in the literature. These are, at least in part, connected to an inverse limit of truncated portions of the de Rham-Witt complex under Frobenius maps. (My understanding of this is that, on the algebro-geometric side, there is some connection to Breuil-Kisin modules, but I understand nothing about these and only know this connection exists from watching a talk of Scholze's.)

2more comments