Brown was originally selected by the Texas Rangers in the first round of the 1986 draft, winning his first major-league start and only appearance of the season. He was once selected to the All-Star team and accrued a 78-64 record over eight seasons with the club. He spent 1995 with the Baltimore Orioles and joined Florida as a free agent after the season.
Brown enjoyed All-Star selections in both of his Florida seasons. In 1996, he led the NL with a 1.89 ERA and a 0.944 WHIP, finishing second in the season-ending vote for the NL Cy Young award and posting a 17-11 record along the way.
On June 10, 1997, Brown came within a hit batter of a perfect game, blanking the San Francisco Giants 9-0 while striking out seven. He completed his Marlins career with a 33-19 record and a 2.30 ERA, striking out 364 batters in his 65 starts for the team.
Brown was dealt to the San Diego Padres in the infamous fire sale after the Marlins took home the world championship following the 1997 season. He later played for the Los Angeles Dodgers and the New York Yankees.
He currently serves as a baseball coach at Tattnall Square Academy.