Differences? Leaving aside the number of folds in puggarees or the fact that the Australian slouch (HAt Khaki Felt Fur) was made from rabbit and other minutiae the principal difference would be that the Gurkha terai is traditionally, at least since the interwar period at least, constructed of two hats, one sewn inside the other. This makes it a somewhat stiffer item than the Aussie slouch.
In miniature, probably no discernible difference, although the top crease usually seems more pronounced on Australian versions, as the material isn't as stiff. Gurkha hats were occasionally turned up.