Higher-Order Components (HOC) and Container Components are different. They have different use cases and solve similar, but different problems.
HOC are like mixins. They are used to compose functionality that the decorated component is aware of.
This is opposed to Container Components that wrap children and allow the children to be dumb (or not aware of the Container's decorated functionality).
It is true when transferring props, that Containers can add functionality to their children. But, this is usually in the form of props being passed down to the children. In Containers, this is also awkward because of the limitation that you cannot simply add props to an already created Element:
So, if you wanted to add a new prop to a child from this.props.children, you would have to use cloneElement. This is not as efficient because it means you have to re-create the elements.
Also, HOC is simply a way (factory) for creating Components. So, this can happen outside the render.
In React, a higher-order component (HOC) is a function that takes a component as an argument and returns a new component that wraps the original component.
A wrapper component, on the other hand, is simply a component that wraps another component.
The main difference between HOCs and wrapper components is that HOCs can inject props into the wrapped component, while wrapper components cannot.