as far as I know, there might be some differences in how the procedure is brought out in each of the federal regions of Germany (small details because there should be a national policy on this matter). I'm talking for intuition. There might be difference in how the services are organized and how much there is available for refugeees depending also on the city. but it's the BAMF deciding on the asylum requests (which is national) i think. Perhaps somebody else in the community can give a more precise answer on this.
I don't think there could be any difference between east and west because nowadays it is not an actual division anymore. East and West were reunified in 1989 with the fall of the Berlin wall.
This is what I would say, as far as I know. I hope it clears some doubts =)
Have a good day and ask away if your have more questions =)