Why do you get prion diseases from eating animal brains but not other parts of the animal?
The notion that you get prion diseases like BSE only by consumption of brain is not correct. Any part of the infected animal that contains nervous system tissue or lymphatic tissue can be infectious, although I guess this is dose-related.

Quote from OIE, "BSE. General disease information sheet" ([1]):


  The risk linked with consumption is addressed by the requirement for
  routine removal of all visible nervous and lymphatic tissue (Specified
  Risk Materials (SRMs)) from carcasses during the processing of
  cattle as well as the removal of any suspect animals from the human
  food chain.
  
  Appropriate handling of nervous and lymphatic tissue tissues from
  BSE-suspect animals in abattoirs and laboratories is indicated to
  avoid accidental exposure (ocular or oronasal). Milk and milk products
  are considered safe.


[1] http://www.oie.int/fileadmin/Home/eng/Media_Center/docs/pdf/Disease_cards/BSE-EN.pdf

The OIE is the animal equivalent of the WHO.