My sincere advice to you. If you encounter something in your life, a practice, a custom, a belief, a tradition etc. just ask yourself or seek if it was there back then at the time of the Prophet (pbuh) or not. Did he used to it or not. Or seek if Qur'an talks about it or not.
Rest, which are not dated back to the exact time of Prophet (pbuh) are all made up things later. The people who follow those traditions simply at max they can do is justify their behavior with that which is there in Islam. Why invent things in the first place? then rotate it and try to prove it right according to Islam. Why? At the first place, what is the need to invent things when the basic things which are out there sufficient and more than enough. Because Prophet Muhammad (pbuh) was a complete way to the truth and the life. Although, those added things maybe right, but why are they needed at all? When the basic things suffice our need.
If you want to stay protected by the creator just remember him, and ask him to protect you. Simple. I can say, it need not be even in Arabic. Believing that God will prioritize you with a Tawiz in Arabic letters of verses in contrast to the one who simply called on God, is... I don't think is a good idea.
So, instead of Questioning why is tawiz Haram, just question, is it damn necessary? Will God protect me only if I wear a Tawiz? Will I be preferred more by God if I wear a tawiz or will I be preferred more if I remember God by saying hundred times "Subhan Allah".
Is it Haram?
Irrespective of whether there are Qur'anic Ayahs in that amulet or not, when Prophet Muhammad (pbuh) said it, he meant it. (I don't have anything extra other than those Hadith which "mkhan3189" provided in his answer)
Why is it Haram?
To this contemporary issue I don't think Prophet Muhammad (pbuh) has explained the reason as why is it Haram. But to guess, it is Haram because there might come a time, suppose when you're protected from something bad, and you believe were protected by 'something', and that 'something' are the pieces of mere papers in your neck written in some Arabic letters which one could hardly read without a microscope. Then and right then, you might not even come close to thinking that it is God who protected you.
And if you believe that you were saved, because of those pieces of paper with some Arabic font, then according to classic Islamic definition it is termed as
Haram 'Shirk'. because the 'Tawiz' protected you, or in other words you believe some material object protected you.
A piece of advice, as the famous and well educated Christian scholars say, "If you know something would lead to sin, then don't play with it or don't taste it at the first place in order to explore how would it feel like, because your fear might come true, because sins are addictive".
May the creator guide us all.