Those two expressions have close meaning:
He is not meant to do this
He is not supposed to do this
What is the difference between them, and when I should use one or the other?
They are freely interchangeable in many everyday contexts, but there are differences in the tone of voice. "Meant to" has a more philosophical, fatalistic connotation, while "supposed to" might refer to a transgression or snafu of some sort. For example:
So, my submission is, the context matters. 'Meant to' might be used in situations that can be remedied as well as in situations that can't, while 'supposed to' might be used in a situation that can possibly be reversed.
As far as what they mean, they are fairly similar. The connotations of the two words, however, differ slightly. "He is not meant to do this" implies a plan or design where the intent is not that he do it. "He is not supposed to do this" implies an expectation or understanding, and seems to come from a more observational standpoint, whereas the first seems to come from a standpoint more knowledgeable about some plan involving the subject and his actions.
Thursagen's answer is right, but I wanted to expand:
1) I think that "meant to" is a bit more formal. "He's supposed to..." strikes as extremely informal and I would keep "supposed to" out of formal writing. Likewise, 'meant to' seems rather off in informal conversation. That said, I think that both of these phrases are rather casual and you would generally use another phrase in writing anyway.
Take this conversation:
Take this piece of writing:
2) "Supposed to" always implies that there exists a definitive set of steps that someone could potentially take to accomplish something. This is not always the case for how "meant to" is used.
You might say:
The relevant meaning is this, I believe:
meaning: It's not his purpose to do this, or he cannot do it.
Example: I am not meant to be a father!
Note that there is a similar sentence:
meaning: He did do it, but accidentally and against his intentions.
This has a very similar meaning as the first, he is not expected or intended to do this, but it can also mean that he is (weakly) forbidden to do "this". It's not strictly illegal to do "this", but hardly acceptable for some reason, like for social norms or an informal agreement.
"Meant to" implies competence. "He is not meant to do this work" (because he's really not cut out for it).
"Supposed to" implies "permission." "He is not supposed to do this" (because he doesn't have permission/approval).
They are used "interchangeably" in the sense of "can." But I consider this misleading.
"Meant" implies that the object was destined for or built for a purpose. "Supposed to" implies flexible intent. The car was meant to go 200 mph but only managed 190 mph. Conversely, I was supposed to meet someone at 10pm, but ended up meeting at 11pm because of a train delay.