Does the "independent life" of an agreement have any other effect after the decree?

Yes, if you want to modify an agreement, whether it is merged or incorporated can have a great deal of significance. An agreement that is merged is a court order; court orders are subject to modification. Therefore, court-ordered support and custody can be subject to modification by the court at a later date. Be aware that if your agreement is "totally merged," even alimony or maintenance becomes modifiable. An independent contract cannot ordinarily be modified except by agreement of both parties to the contract, subject to approval by the state court in which the divorce has been granted.

It may be to your advantage to make modification more difficult, and so you may wish to have the agreement incorporated and not merged.