The main difference is in connotation. Mere is used to prescribe little importance to something. Just can be used in a synonymous manner. For example "I'm merely human" and "I'm just human". This usage of belittles humanity in both cases. I think any differences would be that just might be seen more positively because of its other definitions, like being adequate, or being righteous, while mere is always negative, used to make a person or thing seem small and unimportant.

Sometimes they can be interchanged but quite often they can't. For example, "just" can also mean righteous, like a "just king". It can be a way to express time, like "I just arrived". It implies you arrived shortly before. If you were to say "I merely arrived", that would imply you only arrived and did nothing else - it does not give the reader any sense of time. And so on. (A full list can be found here: http://www.thefreedictionary.com/just.) So basically, "merely" means "only that and nothing more or less". "Just" can have a variety of meaning depending on the sentence it's in.