Yes the W denotes military..
From a purely technical standpoint the tubes ending in GTA, GTB can handle higher voltages (max of 450 volts or 5 watts per section) than a GT can (max 300 volts or 3.5 watts per section). Therefore the GTA/GTB will run cooler.
The GTA/GTB were made for televisions commerically, and for military probably a television type display and other higher voltage applications.
If you want a quiet tube then try a 5692 tube. A 5692 tube is rated for up to 275 volts or 1.75 watts per section.