What the T-38 airplane provides NASA is currency and proficiency in high performance aircraft operations in a jet which has many of the same characteristics as the shuttle in this respect ie high wing loading, high approach and landing speeds, similar glide descent ratios. The White Rocket is a temperamental and challenging little jet to land and will not tolerate sloppiness from her flight crew here. She requires the pilot to stay well ahead of it while making an approach, plan out and execute small corrections very precisely while maintaining airspeed, glide path and angle of attack. Maintaining good proficiency in these aspects of high performance aircraft flight and keeping sharp on that high speed mindset comes in handy in an astronaut's line of work.
And T-38s make excellent chase aircraft, capable of keeping up with the shuttle during the final phases of its descent and approach, and can provide an extra set of Mk 01 eyeballs to verify external functioning of systems, eg landing gear, speedbrakes, hydraulic leaks, etc.
Actually, outside of the shuttle sims, landing approach practice was done in a modified Gulfstream II which included a replica of the shuttle flight deck for the left seat guy. It apparently was remarkably similar to the shuttle in a deadstick landing.