May I recommend the University of Illinois at Urbana-Champaign dissertation "Dislocation Slip and Twinning Stress in Shape Memory Alloys - Theory and Experiments" by Jifeng Wang?
In general, the twinning boundary has lower energy than stacking fault or slip region. Which mechanism of deformation comes into being strongly depends on alloys constituents and structure state. During stress accommodation by slip one or several lattice planes are shifted only by integral numbers of interatomic distances. Slip is realized by moving of dislocations and the lattice in slip region is identical with another lattices in its vicinity.
Accommodation by twinning is realized by moving of partial dislocations in region of twinning plane which means that shifting of atomic layers occur in manner one after one and the shifting of each plane is on distance less than interatomic distance. The atomic bonds are not broken instead of slip. Lattice in twinned region is then mirror image of lattice in untwinned region.
In shape memory alloys there is important parameter called critical resolved shear stress (CRSS). For shape memory effect we need lower CRSS for twinning than for slip nucleation of martensite. For more details I recommend you attached paper.