When Aadam (Adam, may Allaah exalt his mention,) committed his sin, Allaah The Almighty forgave him and sent him down to the earth, but when Satan disobeyed Allaah The Almighty, He did not forgive him and expelled him from His Mercy.
Why were they treated differently?
How did Satan disobey Allaah The Almighty when his actions were predestined?

All perfect praise be to Allaah, the Lord of the worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad, sallallaahu ‘alayhi wa sallam, is His Slave and Messenger.

A Muslim should believe that Allaah The Almighty chooses what He wills and acts as He pleases, and no one is qualified to question His judgment.  Allaah The Almighty Says (what means): {He is not questioned about what He does, but they will be questioned.}[Quran 21:23]  He also is the Utterly Just, Who never commits an injustice even to the measure of an atom's weight.  Allaah The Almighty Says (what means):

•        {And your Lord is not ever unjust to [His] servants.}[Quran 41:46]

•        {Indeed, Allaah does not do injustice, [even] as much as an atom's weight.}[Quran 4:40]

No comparison should be drawn between Aadam and the accursed Satan, since Aadam and his wife turned in repentance to Allaah The Almighty and asked for His forgiveness.  Allaah The Almighty Says (what means): {They said, "Our Lord, we have wronged ourselves, and if You do not forgive us and have mercy upon us, we will surely be among the losers."}[Quran 7:23]  While, on the other hand, Satan refused to prostrate to Aadam and was very arrogant and became one of the disbelievers.  Allaah The Almighty Says (what means): {"O Satan, what prevented you from prostrating to that which I created with My hands? Were you arrogant [then], or were you [already] among the haughty?"}[Quran 38:75]

Even worse, Satan did not admit his disobedience; on the contrary, he insisted on being arrogant and disobedient saying, as mentioned in the Quran, (what means); {"I am better than him. You created me from fire and created him from clay."}[Quran 38:76]  Is there any further arrogance and rebellion against the Beneficent Creator?   Moreover, he went further in arrogance and disbelief and said to Allaah The Almighty as mentioned in the Quran; (what means); {"By your might, I will surely mislead them all, Except, among them, Your chosen servants."}[Quran 38:82, 83]

 Is it acceptable to consider the repentant as equal to the arrogant sinner?

As for your question "How did Satan disobey Allaah The Almighty while his actions were predestined," Satan and his fellow Jinn as well as human beings have the choice to obey or disobey Allaah The Almighty and to believe or disbelieve in Him, that is, they possess the faculties of ability, will, and choice.  Allaah The Almighty Says (what means):

•        {So whoever wills - let him believe; and whoever wills - let him disbelieve.}[Quran 18:29] 

•        {Indeed, We guided him to the way, be he grateful or be he ungrateful.}[Quran 76:3]

•        {For whoever wills among you to take a right course}[Quran 81:28]

•        {He has succeeded who purifies it [Man's soul], And he has failed who instills it [with corruption].}[Quran 91:9, 10]

Therefore, man is responsible for his actions; man obeys, disobeys and believes or disbelieves. If man had not been empowered to choose, no action would have been ascribed to him. Nevertheless, man's will can never take effect unless Allaah The Almighty wills.  Allaah The Almighty Says (what means): {And you do not will except that Allaah wills -- Lord of the worlds.}[Quran 81:29]

Moreover, every person should contemplate on himself and those around him, for instance, how did you write and send this question? How did you choose the words? Did anything force you to write it? Is it not true that you wrote it by your own free choice?

The same thing may happen but on the opposite side, for example, writing something prohibited and sending it. In such a case also, did this happen by free choice or not? 

Consequently, when Aadam disobeyed Allaah The Almighty he did so by his free choice and when he repented, he also did so by his free choice. 

The same thing applies to Satan. That is, when he disobeyed Allaah The Exalted he did so by his free will and he intentionally preferred disobedience to obedience despite the fact that he was called to repent.  Therefore, he rightly deserved to be accursed until the Day of Judgment and he will never find any sympathy in the hearts of the sincere slaves of Allaah The Almighty.

For further information about the issues of whether man's actions are predestined or not, please, see Fatwa 85447.

And Allaah Knows best.