Churches that Teach Tradition as Inspired have Different Traditions. How do They Explain This? Some denominations (Eastern Orthodox, Roman Catholic) teach different Traditions that contradict each other.  An obvious example of this is the papacy (RC says correct.  EO says incorrect.).  Both groups claim to be apostolic.  They may even say each other is apostolic.  Both claim their Traditions to be correct and traceable to God.  How can this be?

Examples of different Tradition.


Nature of bread at Pascha
Filioque
Papacy
Nature of Sin
Nature of Man
Contents of Canon
Nature of authority (council or pope)


See here and here for more details of the differences.

PS.  I'm not asking which of the various Traditions are true.  I'm only asking for an explanation of how those two denominations come up with their Traditions, given that each say their Traditions source to God.
It seems the implicit logic of your question is the following:


Premise 1: group A claim to be infallible 
Premise 2: group B claim to be infallible
Premise 3: doctrine of group A and B contradict each other
Conclusion: neither group A nor group B is infallible


As you can see, this is a non sequitur. The proper conclusion is:


Conclusion: group A or group B cannot be infallible at the same time.


This conclusion still allows either group to be infallible.

So, for the sake of the argumentation, let us assume group A is indeed infallible, but group B is not. How is it possible that group B declares itself to be infallible, without being so?

The answer is evident if we rephrase the question. Is it possible for group B to declare false doctrine? Of course! The alternative - no group can declare false doctrine - is nonsense! In a world were God gives human being liberty, the possibility of error (and heresy) must be allowed. What kind of faith is this if not? 

In fact, the non sequitur you are implicitly assuming destroys every single religious faith in the world. The fact that there are so many different Christian (and Muslim and Jewish, and etc) denominations cannot per se imply that they are all wrong. 

Ask yourself:


  Different Protestant churches derive different (salvific) interpretations from the same Scripture. How do they explain this?