The lady and the tiger both share the similarity of being an unexpected outcome awaiting the alleged criminal on trial in the king's arena. Both outcomes also occur immediately upon opening the door--the man on trial is either instantly killed or married. Both the lady and the tiger represent extremes, of sorts, offering a dire consequence in the fashion of reward or retribution.
The differences are naturally more easy to discern than the similarities. The lady represents a positive reward, and the tiger, a horrible punishment resulting in death. The possible outcomes behind the two doors could not be farther from each other. The lady is all smiles, fairness, and grace, while the tiger is the cruelest, most vicious beast that could be procured.