They're the same. You can construct the geometric realization of a simplicial space $X:\Delta^{op} \to Top$ by taking its co-end with the functor $\Delta \to Top$ which sends the $n$ to the "standard n-simplex" $\Delta^n$. Segal's construction is an explicit description of this co-end in the particular case that $X$ is the enriched nerve of a topological group.
There is something to be said however, since often this construction "isn't right". To be more specific, its really the "FAT geometric realization" which captures the correct homotopy type of a topological group/groupoid, in the sense of the existence of universal principal bundle over it, or, in a more abstract way, it is equal to the weak homotopy type of its corresponding topological stack. Under some mild conditions on your groupoid (e.g. the unit map being a neighborhood deformation retract), then you still get the right answer. That fat geometric realization is essentially what you get by keeping only the strictly increasing arrows in $\Delta$ when you form the co-end.