Application works when OS X was updated but not on a clean install
Filemaker versions prior to 12 may or may not work well on OS 10.10.x

From Filemaker Support:


  FileMaker 12 and earlier versions of FileMaker have not been tested or
  certified with OS X Yosemite (10.10).  These products were developed
  and released before OS X Yosemite became available.
  
  While these earlier versions may install and run on OS X Yosemite
  (10.10), you may encounter installation and stability issues for which
  there is no resolution. (Italics added)


Although you may get older versions (Version 9 was released back in OS X 10.5 days) to work on newer systems, it is hit and miss.  You could try a fresh install again of FileMaker to see if you can get it to 'stick.'  Be careful to keep a backup of your data.  I can imagine if you are still using FM 9 that you have a great deal of important data that has accumulated in your solution over the years.

As for the exact missing files between your installs, that would require a deep look at both installations and compare every tiny little file in the system.  Excruciatingly painful.  It is impossible to determine without access to the system in question.

The files to check are System files in the ~/Library/Application Support/FileMaker and possibly other places.

There are several professional developers who can assist you with migrating your solution to a newer version. (current is FileMaker 14)  This is expensive, but if your business relies on data and downtime is critical, this is a small price to pay.  Converting a file from .fm7 to .fm12 is not trivial, but it is not very difficult for a seasoned developer. (Not assuming your development skill level here.)  This would probably be best as it will prolong the life of your solution.  Especially with OS X 10.11 around the corner.