What is the difference between SLAM and visual odometry?

First, we have to distinguish between SLAM and odometry.

Odometry is a part of SLAM problem. It estimates the agent/robot trajectory incrementally, step after step, measurement after measurement.

new_state=old_state+step_measurement

The next state is the current state plus the incremental change in motion.

Incremental change can be measured using various sensors. Wheel encoders for instance, counts the number of wheel rotation per time step,leading to travelled distance estimation per unit step.

Camera can do odometry as well. Each frame is considered as a step measurement. Moving right or left, up or down indicates a change in the scene from frame to frame. Using computer vision algorithms, the changes could be interpreted to pose change.

On the other side, SLAM is a Simultaneous Localization And Mapping. Localization is a subset of SLAM. Odometry affects localization which in turn affects SLAM.