Biologists affiliated with the Intelligent Design movement nail down the distinction by showing that DNA mutations do not create evolution in any significant sense. Instead, they make birth defects, so the whole thing is false from the get-go. If this is true, that mutations do not lead to more advanced creatures over time, then evolutionary theory, as currently constructed, is false. And the truth, then, would lie elsewhere.
Except that it isn't true. DNA mutations happen all the time -- some are harmful, some might provide an advantage based on the organism's environment but most are simply benign.