I'll give an answer from the old days. The $W$ construction of Boardman and Vogt corresponds to the modern notion of cofibrant replacement of operads. If an operad $\mathcal{C}$ acts on $X$ and $Y$ is homotopy equivalent to $X$, then $W\mathcal{C}$ acts on $Y$. By the way, there is an inherent flaw in the little discs operads $\mathcal{D}_{n}$, namely there is no map of operads $\mathcal{D}_n\longrightarrow \mathcal{D}_{n+1}$ that is compatible with suspension (in the obvious sense: consider $\Omega^n \longrightarrow \Omega^{n+1}\Sigma$). The little $n$-cubes operads do not have this problem, but have others not shared by the $\mathcal{D}_n$. The Steiner operads have all the good properties of both the $\mathcal{C}_n$ and the $\mathcal{D}_n$. In practice, that is in actual applications, such geometric differences are far more important than the questions of cofibrancy and homotopy invariance.

I'll give an answer from the old days. The $W$ construction of Boardman and Vogt corresponds to the modern notion of cofibrant replacement of operads. If an operad $\mathcal{C}$ acts on $X$ and $Y$ is homotopy equivalent to $X$, then $W\mathcal{C}$ acts on $Y$. By the way, there is an inherent flaw in the little discs operads $\mathcal{D}_n$, namely there is no map of operads $\mathcal{D}_n\longrightarrow \mathcal{D}_{n+1}$ that is compatible with suspension (in the obvious sense: consider $\Omega^n \longrightarrow \Omega^{n+1}\Sigma$). The little $n$-cubes operads do not have this problem, but have others not shared by the $\mathcal{D}_n$. The Steiner operads have all the good properties of both the $\mathcal{C}_n$ and the $\mathcal{D}_n$. In practice, that is in actual applications, such geometric differences are far more important than the questions of cofibrancy and homotopy invariance.