Anderson grew up in Bedford-Stuyvesant and Crown Heights, and played jazz early on at the urging of his father, who was a drummer. He played in local clubs from his early teenage years, and studied at the Jazzmobile workshops with Frank Wess, Charles Davis, and Frank Foster. He also met Branford Marsalis, who convinced him to study with Alvin Batiste at Southern University in Louisiana.
Soon after this, Anderson began touring with the Wynton Marsalis Septet, and collaborated with Marsalis through the middle of the 1990s. He continued to play with Marsalis's Lincoln Center Jazz Orchestra beyond this. In 1994, he released his debut album on Atlantic Records; Eric Reed and Ben Wolfe were among those who played as sidemen. His 1998 release Live at the Village Vanguard featured Irvin Mayfield, Steve Kirby, Xavier Davis, and Jaz Sawyer.
Wess is married to former Desimonae Moore of Baton Rouge. They share a son Wessell Anderson IV, affectionately known as Quad. Quad, a University of New Orleans student, who shares his father's passion for music, has now embarked on a music career of his own as an upcoming jazz trombonist. The Andersons have another "baby", a restaurant in East Lansing, Michigan called "Gumbo & Jazz".
- Warmdaddy in the Garden of Swing (Atlantic Records, 1994)
- Ways of Warmdaddy (Atlantic, 1995)
- Live at the Village Vanguard (Leaning House, 1998)