Sometimes it matters a lot. If you are the petitioner you have control of the proceedings and the respondent can only stop the divorce under very limited circumstances and if the petition is based on certain facts.
The respondent can defend the divorce. However, a bare defence that the marriage has not broken down is unlikely to succeed. The respondent can defend and cross petition, that is, the marriage has broken down but it is your fault not mine. Typically a compromise is reached and both parties proceed on their own petition.
What if the petitioner does not bother to apply for decree absolute? The respondent can apply after 3 months but this requires a hearing, is costly, causes delay and is not clear cut.
What of the people who have separated for years, are in new long term relationships but still do not start divorce proceedings – demanding that the other one do so because of the costs? The consequences of not divorcing can be extreme – especially if you have jointly owned assets.
Who petitions has no bearing on the matrimonial finances unless the circumstances are exceptional but it can have an impact on how matters progress. In the vast majority of cases this is not a problem. The unfortunate thing is that when it is a problem it becomes a very big problem!