Rabbet planes are typically very similar to conventional planes except the iron extends all the way to the sides to allow the iron to cut right up to the side of the rabbet, tenon, etc. Even the funny looking ones are pretty conventional in design. As Mike said, they sometimes have a nicker.
Router planes have a very small fence, often resembling a pivot more than a fence. The router plane is designed for cleaning up the bottom of a dado. It is not a great choice for actually making a dado but it excels at cleaning one up. A router plane can follow a curved path. The iron on a router plane extends far beyond the plane body with nothing that can be considered a mouth, a frog, etc. Think of it as a chisel with a depth stop because that's essentially what it is.
The plow plane has a very long fence. It is designed to make grooves down a board, like rails and stiles of a door, and other kinds of dados. It can not follow a curved path, but it excels at making dead straight and flat grooves. On a rabbet plane, the blade comes to the edge of the sides, and you register the plane using the sides. On a plow plane, you instead have a "skate". That's the body of the plane and it is only as thick as the thinnest blade you have (and usually just a hair thinner so things don't bind). The skate and fence provide registration. This allows you to use different sized blades to make different sized grooves.
And then there's shoulder planes. The differences I see between "rabbet" planes and "shoulder" planes seem pretty academic. Some say it's the lack of a nicker, though many rabbet planes lack a nicker. Some say bedding angle. Other say it's bevel up vs bevel down. It all looks the same to me.
I'm not expert on this, but that's just what I've seen to be the main differences.