First of all there's a little weakness in those Ahadith (hadits) which name a number of the names of Allah for example from those 99 Names (Imam at-Tirmidi -who was apparently the only one who narrated this "Hadith" who quotes the names - himself qualifies the Hadith as gharib!) many are neither quoted in the Qur'an nor (elsewhere) in the Sunna -explicitly- as Names of Allah:
- الخافض - المعز المذل - العدل - الجليل - الباعث - المحصي - المبدىء - المعيد - المميت - الواجد - الماجد - الوالي - المقسط - المغني - المانع - الضار النافع - الباقي - الرشيد - الصبور
Now to your question: One important point is that except الله all the names are some kind of characterization which could also be applied on humans in some way. Therefore they are attributes in Arabic and to become a name they need the prefix al-, ar- etc. which is equal to "the" in English so ar-Rahman basically is "the" Rahman etc..
But the Name الله has even in Arabic some specification because it is pure oneness of God (tawhid) in a word itself! As it is significant to الإله = The (one and only) God but more expressive than it, as without the prefix "ال" إلاه has a meaning (God)! And it includes already the article "al" but without it it wouldn't have any meaning! And as it is the name which describes or better summarizes all the characteristics of divinity it's forbidden to name anyone with this name (you can only name abd Allah but not Allah)
ThereforTherefore, in Qur'an even PharaoFir'aun (Pharao) who regarded himself as a God didn't use this name and said
"And Pharaoh said, "O chiefs, I do not know of any other god of yours than myself. Burn for me bricks of clay, O Haman, and build me a high tower so that I may climb it to see the God of Moses for I consider him to be an utter liar.""
وَقَالَ فِرْعَوْنُ يَا أَيُّهَا الْمَلَأُ مَا عَلِمْتُ لَكُم مِّنْ إِلَهٍ غَيْرِي فَأَوْقِدْ لِي يَا هَامَانُ عَلَى الطِّينِ فَاجْعَل لِّي صَرْحًا لَّعَلِّي أَطَّلِعُ إِلَى إِلَهِ مُوسَى وَإِنِّي لَأَظُنُّهُ مِنَ الْكَاذِبِينَ
[Al-Qasas 28:38]"I am your Lord, the highest."
فَقَالَ أَنَا رَبُّكُمُ الأَعْلَى
[Surat an-Naazi'at 79:24]
So when you call Allah in dua' by his name الله you say Ya Allah (O Allah) but when you call him by some characteristic name these names including their articles "al","ar"... obey to grammatical rules and disappear!
By the way if we count all the -different- names the scholars have quoted we find some 276 names
And Allah knows best!