Ok I think I figured out the problem.
There was one entry in the log file for "Faulting application javaw.exe"
Faulting application path: c:\users\[username]\AppData\Local\Temp\l14480385\resource\jre\bin\javaw.exe
Faulting module path: unknown
The installer creates a new temp folder each time it's executed so the number above in the path can be anything.
I verified that the install folder was created in Temp and that javaw.exe existed.
Each time I ran the installer, right before the splash screen loads I would get a "Java(TM) Platform SE binary has stopped working..."
So I cleared out the contents of the Temp folder, rebooted, and the installation worked, loading the splash screen.
There are a lot of web installers that user javaw.exe and leave remnants in the Temp folder.
I'm guessing that when you have multiple instances of javaw.exe in your temp folder, InstallAnywhere has a problem with that.