As far as the outcome is concerned, it should make no difference which party chooses to file the petition. However, one of the parties is able to control the process, the person filing (known as the Petitioner) is ultimately able to control the initial portion of their dissolution case.
Being the Petitioner can provide some advantages:
Firstly, at any proceeding before the court, the Petitioner has the first opportunity to present material to the court. For example, in an evidentiary hearing regarding custody, the Petitioner will be able to outline his or her version of the facts before the other party has the opportunity to do so - thereby shaping the judge's opinion.
Secondly, in order to sever the community, it is essential to file and serve the proceeding - and this may be important if you own a business or earn significant income. After service of the proceeding, the earnings of each party is his or her separate property and the debts of each party are his or her separate property.
Thirdly, at the date of filing and serving your case, the preliminary injunction becomes effective - prohibiting the parties from undertaking certain actions. To the extent that these actions are undertaken prior to the filing and service of the case, there is no such prohibition on the conduct of the parties.
Fourthly, at the time of filing your case, a judge is appointed and thereafter, various motions and requests can be filed with that judge. Until the case is filed, there is no law controlling the parties' actions and there is no judge to whom the parties are accountable.