If one has an understanding of $H_*(LM)$ to begin with, then the bar (or Borel, or Rothenberg-Steenrod) spectral sequence

$$Tor^{\Lambda[\Delta]}_{**}(k, H_*(LM)) \implies H_*^{S^1}(LM)$$

allows you to compute $H_*^{S^1}(LM)$; here the exterior algebra $\Lambda[\Delta] = H_*(S^1)$ uses the group structure on $S^1$. In cyclic homology, there is a nearly identical spectral sequence that's often called "Connes' spectral sequence."

One problem that shows up in computing the $E_2$-term of each of these spectral sequences is how the Batalin-Vilkovisky operator $\Delta$ (or Connes' $B$ operator) coming from the fundamental class of the circle actually acts on $H_*(LM)$ (respectively, the Hochschild cohomology of $C^*(M)$). One advantage to cyclic homology is that $B$ has a very concrete description in terms of Hochschild cochains.

If you can find explicit cocycles which serve as ring generators for $HH^*(C^*(M), C^*(M))$, then you can often compute the action of $B$ directly on these classes. You can then lift this to a computation of the action of $B$ (or $\Delta$) to an action on the entirety of $HH^*(C^*(M), C^*(M))$ (respectively the string topology ring $H_*(LM)$) using the Batalin-Vilkovisky structure on either of these.

My main point is that computing $\Delta$ on low-dimensional classes can be sometimes easier to do using Hochschild homology, and sometimes easier using topology. Then the BV structure allows you to extend these computations to the rest of $H_*(LM)$.