If $F$ is a subexponential d.f. it is well known that the tails of the distributions of the partial sums and partial maxima are asymptotically the same. In this paper we analyse the difference between these two d.f. The main part of the paper is devoted to the asymptotic behavior of $F(x)G(x)-F\ast G(x)$, where $F(x)$ and $G(x)$ are d.f. and where $\ast$ denotes the convolution product. Under various conditions we obtain a variety of O-, o- and exact (asymptotic) estimates for $F(x)G(x)-F\ast G(x)$. Compared to other papers in this field, we don't assume the existence of densities to obtain our estimates.