Just to chime in, in my district's case, Schoology is decidedly NOT a distance learning tool. It is a critical part of our iPad 1-1 in grades 5-8. Teachers upload class materials for students to download to their iPads before or during class, they use the built-in quiz function for quick formative assessments (think exit slips), the discussion feature can serve as a back-channel, just to name a few of the in-class uses. While all of those things (and more) can be done in GApps, either alone or with add-ons, it is the integration of an actual LMS that adds value. All of the features are found in one spot and because we set up Schoology with data from our SIS (PowerSchool), when a student signs in, he or she has easy access to each of his/her classes as does a teacher.
To elaborate just a bit on one feature: In Schoology, a teacher can create a discussion as a stand-alone "object". The discussion starter can be a simple posted question or a linked document or embedded video. The discussion can be opened and closed at specific dates/times (so the teacher can set everything up over the weekend for a discussion that won't occur until later in the week). Different students can be assigned to different discussions (think table groups). Student participation can be scored using a tool that allows the teacher to click and highlight an individual student's contributions and step through each student in turn. While it is possible to use a Google Group for on-line discussions, I don't think it compares to what is possible in the LMS..
Another feature that an LMS can have that would be difficult to pull off in GApps is parent access. In Schoology, a parent creates their own account and then, associate that account with their child using a code we provide. This means we don't have anything to do with managing parent accounts, other than providing the student code. Schoology even has documentation for parents. Once in the system, parents can see upcoming assignments, access the resources posted by teachers, view school news updates. If desired, they can even be allowed to participate directly in courses. It would be possible to run on-line parent education by creating courses and adding parents as "members".
On the practical side, as Bjorn rightly points out, it is possible to build an LMS system from free or inexpensive parts that would be just right for a given school or district, but IMO, there really is no comparison to a system designed from the ground up as an integrated solution. As an analogy: WordPress can be used for a school district website. It's not *really* designed as a full-fledged content management system, but with the right combination of plug-ins and a theme, you can end up with a very satisfactory result. But it'll take a bit of work to set up permissions so that some people can edit some things but not everything. And you'll have to keep on top of all the updates for WordPress and the theme and plug-ins (and hope the various authors don't abandon their products). Perhaps the biggest drawback is that the only one(s) who can support the system is/are the one(s) who built it. While the WordPress community and the plug-in and theme authors can help with issues related to their individual pieces, the knowledge needed to keep the system up and running is often tied up in one or two individuals, and if they leave (or worse)...
Contrast that with using a service dedicated to hosting school websites. Yes, it will likely be more expensive, but it will be designed to support all the workflows typical of a school or district. It will have all the bells and whistles that superintendents and school boards want to see. Different people can be different permissions and staff will have one place to go, one interface to learn, and a single point of contact for support (which wouldn't necessarily have to be you!). And if you hit the PowerBall some weekend, the folks you leave behind on Monday will have a phone number to call to keep things moving.
In the past, when I had more time, I was very much a DIY person when it came to systems. In many cases, that was because the solution I was after didn't exist and frankly, I relished the challenge. As time has passed, my priorities have changed, and with the availability of solutions more sophisticated and polished than anything I could create, I do not hesitate to recommend spending money for critical pieces of learning infrastructure.
-Tom