Purpose of hiring Klinsmann is different
I believe this is one of the reasons. Before Klinsmann took over Germany international team, they were running in circle and the atmoshpere was dead. So what Klinsmann did there was to change. He brought the new thoughts and tactics to the team so his change worked. Klinsmann brought a lot of new young players into the national team and made them from an old and dying team to young and energetic team. This was what Klinsmann did very well. In Bayern, the purpose was simple - winning. Franz Beckenbauer even wanted him to change things around in order to achieve the European success but he had bottom line - Bundesliga must be in their pocket. So Klinsmann could not change the team. Now Bayern is even in trouble for protecting the place of Champions League so I believe this is the reason why Klinsmann needed to go now. Under this difference, I believe this is one of the reasons why Klinsman failed to do his job in Bayern.
The team culture was very different in Germany national team and Bayern. When Klinsmann took over the Germany national team, he also brought Joachim Löw and Oliver Bierhoff to assist him, plus the upper management were open and willing to accept the change, so Klinsmann could change the team from the bottom. Also, the upper management of national team were dared to follow his change and gave him confident, even after a humiliating 4-1 loss to Italy in friendly. However, Bayern was different. The culture was too conservative that even Franz Beckenbauer accept the new thoughts to pump into the team, Klinsmann must follow his way. Klinsmann had too much pressure in Bayern internally so he could not make the team perform well externally. Furthermore, Klinsmann did not have much authority in the operational issues in the team, so he was very hard to get his thought through to the team and led to massive failure.
Time of preparation
This is one of the biggest differences between taking over in Germany national team and Bayern. As Germany national team had 2 years to prepare for the World Cup in their home turf, so Klinsmann could maximize the experiment on the 2-year long friendlies. As mentioned before, Germany had suffered a humiliating 4-1 defeat against Italy under Klinsmann control, I believe this was just his another experiment. The Confederation Cup in 2005 was only an experiment only, so even though there were mistakes after mistakes, it was not a concern, as Klinsmann knew the team was growing and World Cup was his real deal, the other matches were not his concern. However, Klinsmann only had 1 month to prepare the big matches in Bayern, as every match was important, he did not have much time to prepare and implement his plan. As he did try to change the formation to 3-5-2 in the start of the season, he had to go back to 4-4-2 before he saw any good result in his ideal formation. This was the evidence that Klinsmann basically did not have any spare time to prepare and led to failure.
Coaching team difference
Klinsmann was not the only person who took the pride for bringing Germany national team from the dark tunnel to bright freeway, Oliver Bierhoff and Joachim Löw were also important. Löw, who even had a limited experience on coaching, he could effectively implement Klinsmann's plan to the real football pitch. Löw in fact was the most important person in Germany national team as he could bring the thoughts from Klinsmann into practical way. This was the reason why Löw took over the national team job and continue the good run in Germany national team. Oppositely, I believe the biggest failure of Klinsmann was hiring his American coaching team. Martin Vazquez was not the appropriate person to assist Klinsmann, even Vazquez even could not speak German. So Klinsmann's thoughts basically could not convert into the football pitch. As Philipp Lahm complained before, Klinsmann only encouraged the team but he could not bring out any practical solution when the team was trailing or in the deadlock. This is the failure of Klinsmann and he could not deny it.
Summing up Klinsmann is the person who can bring out the long term successful plan to the team, as his thoughts and strategies really did help the team to bring out from trouble. However, what he needs to learn more is the real match situation. It is very dynamic in the football pitch and Klinsmann must think faster. On the other hand, he is very good at choosing and encouraging players as this is very important to any teams in the world. His enthusiasm can lead the team to do the impossibles, this is what he does best. So far he needs an experienced assistant, not another "just" American coach to assist him in the real pitch situation. When Klinsmann gains much experiences, he will be one of the best coach in the world.
All in all, wish Klinsmann, my forever idol, all the best!!