A symptom gives us evidence for something. A high temperature may be a symptom of infection — it typically accompanies it, and its presence makes infection more likely. But a criterion is different: it is what we call a certain condition. The criterion for angina, for instance, might be a particular bacillus; its presence doesn't merely indicate angina, it is what angina consists in according to our current medical practice. Criteria are internal to our conceptual scheme in a way that symptoms are not.
The philosophical importance of the distinction is most visible in discussions of mind. When we say someone is in pain, what is our evidence — and is that evidence criterial or symptomatic? Wittgenstein argues that pain behaviour — writhing, groaning, guarding a wound — functions as a criterion, not merely a symptom, of pain. It is not that behaviour gives us fallible evidence of an inner state; the concept of pain is at home precisely in the context of such behaviour. This does not mean that people cannot pretend, or that pain just is behaviour — but it changes what kind of knowledge claims about others' pain amount to.
One of Wittgenstein's subtler observations is that criteria can shift. What serves as the criterion for a concept in one context may become merely symptomatic in another, as theoretical understanding deepens. This is not a defect of language but a feature of how our conceptual practices develop with our practical and theoretical needs. Philosophy's mistake is to freeze a criterion and demand its universal application — to ask "but what is pain really?" as if one answer must hold across all contexts.
The criterion/symptom distinction is developed in the Blue Book (pp. 24–25 of the standard edition) and takes on a central role in discussions of psychological concepts in the Investigations. The concept is analyzed at length by Gordon Baker and P.M.S. Hacker in their multi-volume commentary.