Self-serving bias and actor-observer bias are both types of cognitive bias, and more specifically, attribution bias. Although they both occur when we try to explain behavior, they are also quite different.
Self-serving bias refers to how we explain our behavior depending on whether the outcome of our behavior is positive or negative. For example, an athlete is more likely to attribute a good performance on their own ability, and a poor one on external causes like the event environment.
Actor-observer bias refers to how we explain the causes of (undesirable) behavior. When we are the actors, we attribute our behavior to external factors, while when we are the observers we are more likely to attribute the same behavior to internal factors. For example, when we drive dangerously, we may attribute this to the poor visibility on the road, while when another driver exhibits the same behavior, we are more likely to think they are just bad drivers.