QT won't open it either.
QuickTime only knows MPEG-4 AAC bitstreams in the MP4 container, no other file formats like ADTS AAC with the *.aac file extension.
I converted it from a .mp4 file using dBpowerAMP, using the codec they offered to do this very thing.
dBpowerAMP uses mp4creator for extracting AAC files from the MP4 container or vice versa. Other Windows GUI applications with this option would be foobar2000 or Speek's "Ivan & Menno". This is not a transcoding process, by the way, just an extraction without changing the sound.
Is there some better way to convert the file from .mp4 (or .m4a) to .aac? Do I need/want to convert to .aac?
You could also use mp4creator on the command line, but usually users prefer a GUI for it, e.g. mp4UI. And you only need *.aac files if your hardware does not support *.mp4 or *.m4a files like some Nokia mobile phones, the DivaGEM or some Philips Expanium CD portables.
Does it need that extension to work on the iPod? Is it any smaller?
The iPod only knows *.m4a, *.m4b and *.m4p files, and *.aac files are a little bit larger due to the ADTS headers for each frame. There's no sound difference between them though.