Hi Noah,
The first thing to say is that you can't have bodies outside of the component, not in Fusion. When you starting your design you are already inside the component. When you hit New Component button Fusion creates subcomponent. Then you can create other subcomponents or a subcomponent of that subcomponent:

To answer your initial questions "What can bodies do that components cannot?", with bodies, you can do geometry with components you can't. For components to have any geometry you need to create a body within it.
It may be easier to understand if I would make reference to another cad tool. Components are, in fact, like a Part files (SW or ProE) by default they contain origin and basic planes and axis. As it is in other programs to create geometry we are creating bodies. In Fusion if you don't create a new component, whole design will work like a Part file. That component/Part file can contain multiple bodies. If on other hand you would create a new component, our top level component will serve also as an Assembly file and all new components will be like Part files/subcomponents. Of course, these subcomponents may contain more subcomponents, then we can treat our file like an assembly full of subassemblies.
All that could be a little confusing. We can reverse your question, What can components do that bodies cannot? You can't assembly bodies. If in your design you have only bodies and no new components you have no Parts to assembly.
Another important thing to say is that through all that time you can edit one component in a context of others, you can create multiple cross references.
It's like you would be in an assembly in Solidworks and could edit Parts inside, in a context of that assembly.
It is a very powerful concept where whole structure is like matryoshka toy. Each component is a Part file and Assembly file in the same time, each component is built in the same way.

That creates one problem you can have bodies and components on the same level. To address that "inconsistency" and couple of others following consequences Peter (@TrippyLighting) is proposing workflow where you treat your First top level component as Assembly file only. That require slightly more constraint/focused workflow because you need remember about activating components and making cross-references wisely.
You can also try to go old fashion and treat each design as a Part or Assembly file only. Then in "Part" designs you would stick to bodies only and creation/modification tools in "Assembly" designs you would assemble "Part" design with the use of linked components, and you would use only mating tools (joints).