People will not be punished for sins they were forgiven, as can be seen in many verses:
Their (disbelivers/sinners) requital is that there shall be upon them the curse of Allah, the angels, and all mankind. They will remain in it [forever], and their punishment shall not be lightened, nor will they be granted any respite, except those who repent after that and make amends, for Allah is all-forgiving, all-merciful. (3:87-89)
You have no hand in the matter, whether He accepts their repentance or punishes them, for they are indeed wrongdoers. (3:128)
Indeed the hypocrites will be in the lowest reach of the Fire, and you will never find any helper for them, except for those who repent and reform, and hold fast to Allah and dedicate their religion [exclusively] to Allah. Those are with the faithful, and soon Allah will give the faithful a great reward. (4:146)
and there are many more such examples. The thing to see here is the dichotomy established with the 'except's and 'or's between the two things: either you repent or you are punished. Meaning repentance saves you from punishment. So keeping with Allah's quality as The Most Merciful and the ayaat of the Quran, it can be said that forgiveness saves people from punishment.
More ways of showing:
Every one of the children of Adam has been created with three hundred and sixty joints; so he who declares the Glory of Allah, praises Allah, declares Allah to be One, Glorifies Allah, and seeks forgiveness from Allah, and removes stone, or thorn, or bone from people's path, and enjoins what is good and forbids from evil, to the number of those three hundred and sixty joints, will walk that day having saved himself from the Fire. (Muslim)
Notice how seeking forgiveness and other things are directly linked to saving people from the Fire (punishment).
The word for forgiveness in arabic (غفر) signifies covering/hiding something or pardoning punishment. Neither meanings are fulfilled if Allah punishes people who he has forgiven. He would neither be covering up the sins as the punishment would be obvious for everyone to see and nor would He be really pardoning punishment as He would still be punishing.