Suppose we have a ring spectrum R with multiplication $\mu$ that also has a diagonal map $\Delta$ (for example the sphere spectrum). The cohomology $R^* R$ has a ring structure, like it would for any spectrum, coming from the fact that we can define the product of $f: R \rightarrow \Sigma^n R$ and $g: R \rightarrow \Sigma^m R$ to be $(\Sigma^m f) \circ g$.
It is also the case that a ring spectrum endows the cohomology of a spectrum with a diagonal with a (non-unital?) ring structure. This is given by taking the product of $f': E \rightarrow \Sigma^n R$ and $g': E \rightarrow \Sigma^m R$ (with domain having a diagonal $\Delta '$) to be $\Sigma^{n+m} (\mu) \circ (f \wedge g) \circ \Delta'$.
How do these different ring structures on $R^* R$ interact? Again, taking the sphere spectrum, these coincide because $\mu = \Delta = Id$ and $f \wedge g= f \wedge Id \circ Id \wedge g= (\Sigma ^{m} f) \circ g$.
I don't think there's any particular relationship between them in general. For instance, suppose $R$ is $\Sigma^\infty_+ M$ for some discrete finite monoid $M$. Then the first ring structure on $R^0R$ is just the ring of endomorphisms of $\mathbb{Z}[M]$ as an abelian group, which is just a matrix ring $M_n(\mathbb{Z})$ where $n=|M|$. On the other hand, the second ring structure is $\mathbb{Z}[M]^n$, a product of $n$ copies of the monoid ring. These two ring structures are never the same if $n>1$, and I don't see any nice way they would interact.