Both are ways of adding and pushing/pulling vertices and faces to create a shape. It is possible to get the exact same results via either method, but like all tools, each one is better for certain jobs.
Your second sentence is generally correct, but do keep in mind that each one is usually better, not always better. It's also up to preference.
You do not HAVE to sculpt your characters, although it may be the best way to get the results you want. Other objects can be sculpted as well. It's all about the best way to get the end result.
In regards to the process you mentioned, that is a very common way of creating things like characters. The reason for doing it this way is because sculpting usually results in objects with high amounts of faces and terrible topology. Blender's Dyntopo (Dynamic Topology) system is often used to sculpt, but it created a mesh entirely made of triangles. Triangles are generally avoided if possible, as they can cause deformation issues when animating (usually when combined with subdividing). A dense triangle mesh is going to animate a whole lot worse than a mesh that has nice topology and is composed entirely of quads. That is why the retopology process is used. You take a detailed model that is the final shape, but that wouldn't deform and animate well, and recreate it by placing the faces where you want them so that your model will deform well (there are a number of ways and tools to do this, but they all involve ways of 'sticking' the new faces on the sculpted model so that you simply have to restructure your model, not recreate it).
You can still sculpt on a mesh that has been modeled, and that is a common way of adding detail to a mesh while still having clean topology, as at that stage the multiresulution modifier would be used instead of Dyntopo, and the modifier keeps the topology by subdividing the mesh and allowing you to sculpt the subdivisions.