The difference between a risk matrix and a risk register is that:

A risk register is a comprehensive document that provides detailed information for each identified risk, including its description, likelihood, impact, mitigation strategies, and crucially, the designated risk owner responsible for monitoring and managing it. This ownership detail is a fundamental component of a risk register that is not typically present in a high-level risk matrix.

A risk register typically contains significantly more detailed information than a risk matrix, which primarily focuses on plotting risks by likelihood and impact.

A risk matrix visually plots risks based on their likelihood and impact but does not explicitly state or document the risk owner as one of its primary functions.

Both a risk matrix (typically in conjunction with a register or as labels) and a risk register list risk descriptions, so this is not a distinguishing difference between the two.