Structure of carbon fiber
The key difference between graphite and carbon fibers is the degree (or range) of ordering.  In graphite there are large, extensive planes of carbon stacked one atop another resulting in long-range order. In the fibers, there are small ribbons of graphite-like (it is not all carbon, due to the way the fibers are prepared there is still some nitrogen present in the molecular array) material oriented along the fiber axis and there is only short range order.  As a consequence graphite with its extensive 3 dimensional planar array of ordered carbon atoms is crystalline.  Whereas the fibers, being ordered over a much smaller range,  are amorphous.  The amorphous nature of the fiber allows the smaller ribbons to bend and fold, and mechanically interlock with one another.  This mechanical interlocking is what serves to increase the fiber's strength.  See here for a very nice and concise description of how the fibers are prepared.  The pictures in the link make clear how the short range ordering in the fibers results in a highly folded and interlocked structure.  Further heating of the fiber will convert it to graphite.