The most obvious connection between Tom and Gatsby is Daisy. Tom is married to Daisy, but shows little affection towards her. Gatsby is madly in love with her despite the fact that she is a married woman. Tom buys an apartment in the city to get away from Daisy with his mistress. Gatsby buys a mansion across the harbor to be close to Daisy. He throws elaborate parties in the hopes that she will wander in.
Another connection between the two is the death of Gatsby. Tom's mistress is killed in a car accident and Tom convinces her husband that it was Gatsby. He persuades her husband that Gatsby was her lover instead of him. Her husband then goes looking for revenge and kills Gatsby.