Similarities: health and longevity are primary goals of both tai chi and hatha yoga, and many people cross-train in both disciplines. Both emphasize stretching, breath work and chi, called prana in yoga. Both seek to integrate the body, subtle energy, and mind. Their approaches to physicality are significantly more subtle than ordinary Western exercise methods and sports. Both help calm your mind and deal with stress.
Differences: here is a very simple way to explain the difference: in tai chi, you relax to stretch; in yoga, you stretch to relax. Tai chi emphasizes stretching through sophisticated dynamic fluid motions rather than by holding static postures. Yoga tends to use more extreme stretches than tai chi and some postures lock the joints and arch the back, which never happens in tai chi. These poses can be difficult for those with back or joint problems.
This tai chi training was a tranformational experience, breaking through one's sense of expectations, releasing stagnant areas in the body, mind and spirit.