There might be a requirement to invoke the Javascript method in the child component from the parent component and in such scenarios <aura:method /> comes handy.
Aura framework is a component-based framework and when we break a huge functionality into small pieces the major challenge we might come across is sharing of the resources.
That's where <aura:method/> comes into the picture. I would say this feature is an absolute piece of beauty in the complete Aura Components framework. The flexibility it provides is next to none.
It solves one exclusive problem which is being able to invoke the JavaScript method in the child component from the parent component.
Here is the snippet of code displaying it.
<aura:method /> is exactly similar to @api decorator in LWC. And this is the exact opposite of Component Events.