Machine learning-based statistical AI is primarily concerned with “inductive” thought: given a series of patterns, infer the trend. Classical AI, on the other hand, is primarily concerned with “deductive” reasoning: derive a conclusion from a set of constraints. Another distinction, as noted in the preceding question, is that C++ is a popular language for statistical AI, but LISP is the language of choice for classical AI.
A system cannot be genuinely intelligent until it exhibits both inductive and deductive thinking capabilities. Many people feel that, in the end, there will be some type of statistical and classical AI synthesis.