The fact that you're asking this question means that the answer is no, you do not need application state management. I don't mean that in a rude way, but even Dan Abramov (the creator of Redux) thinks that you might not need Redux. Basically, if you don't come across the pain point the Redux solves, then don't use it.
Having said that, I've used Redux (please note I'm not an expert at it) and I find it useful for the following reasons:
There are other reasons that Dan points out in his article, but for me I think it makes sense simply from an organizational sense. Having everything also separated out and very "strict" means that there's less room for error, which is crucial for large teams working on a large app. However, Redux doesn't tell you how to write your app. I am able to reduce a lot of boilerplate by using redux-actions, so my entire state is kept in a single file instead of having my constants, actions, and reducers all over the place.
To re-iterate my initial point though: if you don't think that you need it, you probably don't. Chances are you're not building a Facebook so your app will do just fine with local state or whatever you're currently doing.
Atul Sharma
Full Stack Developer | Cloud Native Applications