They are different ideas. "Mess" means untidiness: lack of organisation. All of the things in your room are important and necessary, but you need to tidy them up and put them away in some kind of order. "Clutter" means that there is too much stuff. You can't find anything in your room because there are too many things there. You don't only need to tidy your room - you need to call Marie Kondo and sort through your belongings to decide which things you need to throw away.

I agree 100% with Su. Ki.'s explanations.

Hmm, they are fundamentally different concepts like the first person explained, but in the way that you mentioned, they are interchangeable sometimes, when you're referring to physical objects. And thinking about intangible things like the mind, I realised that you can ALSO use both! Like "my mind is really cluttered" or "my thoughts are in a mess". But there are situations where you can only use "mess", like "the organisation of the project was messy" but you can't say that it's cluttered.