Weighted network congestion games are used for modeling interactions involving finitely many non-identical users of network resources, such as road segments or communication links. In spite of their special form, these games are not fundamentally special: every finite game can be represented as a weighted network congestion game. The same is true for the class of unweighted network congestion games with player-specific costs, in which the players differ in their cost functions rather than their weights. The intersection of the two classes consists of the unweighted network congestion games. These games are special: a finite game can be represented in this form if and only if it is an exact potential game.