Correction: It's normal to be left-handed, but it's not the norm.The term "not normal" has no negative implications for me. It's just 'not normal' to be left-handed or have extremely high or extremely low an IQ.
No matter what Merriam-Webster may say, and no matter what you may think is logical, in contemporary American English, both "abnormal" and "not normal" are very far from "neutral" sounding, and in many contexts can be offensive if used carelessly. (For example, if a US politician or other public figure were to say "Children with Down Syndrome are abnormal," he'd be burned in effigy, or at the very least, millions of people would be waiting in line to throw dog sh*t at him.)
On the other hand, "the norm" and "not the norm" are very neutral, and thus much "safer" phrases if you want to say "like the majority" or "different from the majority."