Legally? None. Emotionally? A ton.
Illinois is a no-fault divorce state. In addition to allowing divorce based on “irreconcilable differences,” this also means that adultery will not affect property settlements. Property settlements are to be made “without regard to marital misconduct.”
In some states, adultery can affect alimony awards. For example, in Virginia, proof of adultery could possibly lead to a complete bar from receiving alimony. Fortunately or not, that rule does not apply here in Illinois. Instead, alimony is also awarded "without regard to marital misconduct."
A wandering spouse will also usually not affect child custody. The relevant statute states, "[t]he court shall not consider conduct of a present or proposed custodian that does not affect his relationship to the child." This may change however, if the other parent can show that the adulterous parent's activities are affecting his or her relationship to the child, such as a relationship with an abusive or alcoholic paramour. Instead of worrying about the parents' social lives, the court determines custody based on the best interests of the child.
Where it does come into play is both in terms of peace of mind and as fuel for further argument. For some people, having it legally determined that it was the other spouse's fault is necessary in terms of emotional closure. For others, it's merely an excuse to fight the spouse over another of the many issues that have plagued the marriage and led to its demise.
A good Illinois divorce attorney will hopefully be able to tell the difference between the two and inform their client as to the legal irrelevance of an adultery-based divorce.