Historically all states in the U.S. required one spouse to prove that the other spouse was legally at fault to obtain a divorce. Fault grounds included adultery, cruelty, desertion, confinement in prison, physical incapacity, and incurable insanity. This led to a number of complications in the divorce system, wherein spouses would seek ways to make the court grant a divorce, such as contrive to have the other spouse cheat. In 1969, California became the first state to grant "no-fault" divorces. Today, many states don't consider fault at all when granting a divorce, and all states have at least a partial "no-fault" divorce option. Proving adultery or infidelity in marriage in order to be granted a divorce has become somewhat moot.
In some cases, however, fault grounds may be sought. This is more common in states such as New York, where there is a long separation period required before a "no-fault" divorce is granted. In these cases, proving adultery can get the waiting period waived and result in a more rapid dissolution. Also, a party may try to prove adultery in order to gain more beneficial parenting or spousal maintenance (alimony) orders. States are varied as to how proving adultery affects those orders, and, for the most part, don't take it into strong consideration.
There may also be instances where it is beneficial for one spouse to prove adultery, such as in cases where a prenuptial agreement is signed. In such cases, often one spouse has to give up more in terms of payment or custody if it is proven that they have been unfaithful. This is not guaranteed however, and it is often extremely hard to prove infidelity. It usually takes more than a simple admittance by the adulterous party.
Since divorce laws vary from state to state, you will have to do research to see if your state of residence recognizes any advantage to proving adultery in divorce proceedings.
Dissolution laws in Washington State (http://www.wsba.org/media/publications/pamphlets/dissolution.htm)
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