$\require{cancel}$
Recently I have witnessed the proof of the fact that if $R$ is noetherian so are $R[X]$ and $R[[X]].$
From what I read on wiki I guess that if $R$ is $\cancel{\text{local ring}}^*$/integral domain so are $R[X]$ and $R[[X]].$
Can you cook up some property of $R,$ such that the ring of polynomials $R[X]$ also has that property and the ring of formal power series $R[[X]]$ does not?
Note. I am not looking for differences between $R[X]$ and $R[[X]]$ which hold for arbitrary commutative ring $R.$
EDIT$^*$. user26857 pointed out that $R[X]$ do not have to be local if $R$ is local. Clearly for every point $a\in k^n$ ideal $(x_1-a_1,\ldots,x_n-a_n)$ is maximal in $k[X]$.
One standard example is that if $R$ is a UFD, $R[X]$ is a UFD, but $R[[X]]$ may not be. Such examples can be found in P. Samuel's lecture notes on UFD, TIFR lecture notes. See the chapter on power series over factorial rings, corollary 1.