The difference between these two is that racism is an attitude while discrimination is an action. The two can go together, but it is not necessary that they do.
Racism is the attitude that someone else is inferior because of their race. You do not have to act on this attitude; you just have to feel it. By contrast, discrimination is when you treat someone differently because of their race. You do not have to feel that they are inferior. For example, if you worked in a place that would not serve African Americans as a matter of policy, you could feel that African Americans were equal to whites (and therefore hate the policy) but keep on discriminating so as to keep your job.
Racism is a feeling, discrimination is an action.