While there are no set-in-stone advantages of being the spouse that initiates the divorce by filing in court, the inherent nature of being the filer, as opposed to the responder, is what makes it the more desirable position of the two.
By filing, it can be presumed that you have met with your attorney, and have a basic idea of what you will be seeking in the divorce, besides the termination of the marriage itself. In contrast, the spouse who is receiving the divorce notification then has to find an attorney and file a response. You also have the choice of venue, meaning where the case will be heard, which is a luxury considering that, if the roles were reversed, you may have to appear in the state you formally resided in with your spouse even if you had just recently moved upon separating.
The petitioner, or the spouse that files, is granted the opportunity to make their initial argument first at trial. Again, while this is not always the case, this can potentially make an impression on the court that influences the outcome of the overall divorce, however slightly.
The real advantage of filing first is that you will have had the time to meet with multiple attorneys, and thoroughly vet them in order to find the one you feel most confident in having represent your interests. Once you make that decision, you still have the opportunity to formulate a loose plan of attack before going into the divorce process – a luxury the respondent will not have.