Name one thing a believer can do that a non believer cannot?
1: Believe!
The carnal mind is enmity with God (enmity: a feeling or condition of hostility; hatred; ill will; animosity; antagonism).
Rom 8:6 For to be carnally minded is death; but to be spiritually minded is life and peace.
Rom 8:7 Because the carnal mind is enmity against God: for it is not subject to the law of God, neither indeed can be.
Rom 8:8 So then they that are in the flesh cannot please God.
We can be as good as we can be -but we can not be as good as we need to be to live forever -or even understand that we need to be better -without God.
Humans can be good. Basic good and evil is apparent in what can be seen -and some parts of God's law are obvious and can be kept by those who have never heard the law (ten commandments and applicable statutes and judgments).
Rom 1:19 Because that which may be known of God is manifest in them; for God hath shewed it unto them.
Rom 1:20 For the invisible things of him from the creation of the world are clearly seen, being understood by the things that are made, even his eternal power and Godhead; so that they are without excuse:
Rom 2:26 Therefore if the uncircumcision keep the righteousness of the law, shall not his uncircumcision be counted for circumcision?
Rom 2:27 And shall not uncircumcision which is by nature, if it fulfil the law, judge thee, who by the letter and circumcision dost transgress the law?
Rom 2:28 For he is not a Jew, which is one outwardly; neither is that circumcision, which is outward in the flesh:
Rom 2:29 But he is a Jew, which is one inwardly; and circumcision is that of the heart, in the spirit, and not in the letter; whose praise is not of men, but of God.
We see good all around us, but we see much evil, also -and the good are not completely good -nor can they bring peace to the earth.
The most significant thing a non-believer is unable to do is to "believe" -and if one does not believe, they cannot live forever. Everyone will be given an opportunity to believe -some even after death, but the fact remains. Many also believe that which is not true about God.
Most can see sense in a few of the ten commandments, but many find some of them quite ridiculous. If they do not believe in God, why would they obey him before any other, take his name seriously, keep the sabbath, etc...
Still -believing is not enough. After one is truly called by God (God always initiates this with those who are TRULY called -all will be eventually, but not all before Christ's return -those called from Adam until Christ's return are called "firstfruits" -many who have lived will be given eternal life later), and responds by beginning to obey HIS commandments (repenting of sin), they are to be baptized and receive God's holy spirit within them after the laying on of hands of one of his ministers. Many misunderstand this holy spirit -thinking it is a separate being, but the bible definition is as follows....
2Ti 1:6 Wherefore I put thee in remembrance that thou stir up the gift of God, which is in thee by the putting on of my hands.
2Ti 1:7 For God hath not given us the spirit of fear; but of power, and of love, and of a sound mind.
It enables one to become more like God in many ways -to see from his perspective and even to use spiritual "gifts". It enables one to see their own sin more clearly -to begin to keep the commandments in the letter and also the spirit of the law -and to see why they are so important to a greater degree.
Humans can do good -but they cannot do what needs to be done in order to bring peace to earth. God can and will -quite soon, actually -and many non-believers find this ridiculous.