Most series networks that you see are first order series networks. There are some advantages in that with fewer parts you can actually get more of a second order roll off but with fewer parts.
There can also be some advantages in the phase relationship of the drivers.
The disadvantages are that any change to one value effects both drivers. And very few drivers can be used with a first order series filter designs. At least drivers involving a woofer and a tweeter.
Series filters that I have used have been higher order series filters. If you understand them then it is no harder than designing any other filter. Most people however just don't get it. It isn't something you can just calculate either. You need all the tools of a good measuring system to make it happen.
And having A/B'ed the same speaker using both series and parallel networks, that have both been optimized for that design, doesn't really show much of an audible advantage either way (in my experience).