Wine is forbidden for a Muslim here on earth:

See also Sura 2:219.

On the other hand in Paradise are rivers of wine:

Is wine good or bad? Are we forbidden on earth something that is truly good? Or in Paradise are we not only allowed it, but even provided in overflowing measures (rivers of ...) something that is so bad that it is called "Satan's handiwork"?

Some Muslims attempt to explain away the discrepancy by appealing to the following passage:

The heavenly wine, it is claimed, which believers will drink will not cause headaches or madness, thereby justifying Allah's decision to permit it in paradise.

The main problem with citing this is that the passage nowhere identifies this drink as wine. It does not even mention either the word wine or strong drinks/intoxicants. It simply says that believers will drink from a cup something which is white in color which does not cause headiness. If anything, one can make an argument that this drink is actually milk. After all, the Quran does mention the fact that believers will be given rivers of milk, and milk is indeed white:

Here is the other text which Muslims quote to prove that the wine in paradise will be different:

Again, the reference nowhere mentions wine and can be referring to the other drinks which the Quran says will be given to believers, i.e. honey, milk, water etc. In fact, the Sura goes on to mention the flowing waters which believers will have:

Thus, one can legitimately say that the spring which the Quran says will not cause throbbing or intoxication are these very waters. This becomes even more apparent when we realize that the Quran mentions the fruits that the believers will eat from right after the Quran refers to the spring and flowing waters:

In light of the foregoing, it seems pretty certain from the context that the drink which the Quran says doesn't cause intoxication is in fact the very flowing waters that are mentioned later on in the chapter.

Moreover, one can even argue that the reason these passages emphasize that these particular drinks don't cause headiness or intoxication is to distinguish them from the wine which believers will have access to when in paradise. In other words, the qualification presupposes that these drinks are different from the wine which believers will be drinking since, unlike wine, these specific drinks don't cause madness or headaches. Putting it in another way, these texts are not identifying these drinks as wine, but are actually contrasting them to wine by highlighting the point that they don't have the common affects associated with intoxicants.