Is it a condition that the servant not be punished on the Day of Resurrection if he repents of a sin and Allah forgives him in this world? And what is the difference between the state of those who commit sins and those who commit them openly, in terms of forgiveness, repentance, and punishment?

If a servant repents with sincere repentance, Allah accepts his repentance and forgives his sin, and he is not punished for it, neither in this world nor in the hereafter. So, the one who repents is not tormented. The difference between those who commit sins discreetly and those who commit them openly is that the open sinners are more exposed to Allah's wrath, as stated in the Hadith: "All of my Ummah will be forgiven except al-mujahirin (those who commit sins openly)." If an open sinner repents with sincere repentance, Allah accepts his repentance and forgives all his sins.