To take your last question first... "If one can have a subjective prior for anything, then what's the point of differentiating between claims?" There are convergence theorems in Bayesian theory that show that if two people start from different priors and update those priors by Bayesian conditioning on a sufficiently large quantity of independent evidence, then their posteriors will converge. The result is rather idealised, because if the two priors are a long way apart then convergence may not be feasible in practice. But the 'point' of the Bayesian approach is that updating one's beliefs based on evidence is better than not doing so, and people who initially disagree may come to agree when presented with sufficient evidence.
"Why should a high prior for God for example be seen as rational as a low prior?" The rationality of the prior is not the issue in subjective Bayesianism. Your priors are just whatever they are. The rationality lies in the use of Bayesian conditioning as an updating mechanism. At the end of the day, your priors are just a matter of what seems plausible to you, and what seems plausible to you may not seem plausible to others.
"What's the difference between an unfalsifiable but wrong theory and a theory that just hasn't been confirmed yet?" An unfalsifiable theory has no practical use. A theory that is in principle falsifiable but has never been tested yet might be useful if it can be corroborated by surviving lots of testing.