Does it make a difference as to who divorces whom?

In the majority of cases it should have no impact on the eventual outcome as to who starts the divorce process. 

There may, however, be financial implications as the Petitioner (who issues the petition) will incur additional costs but depending on the grounds of the divorce, may seek to recover  those costs from the Respondent. (This relates purely to the costs of the divorce and not any costs in relation to resolving for instance the financial aspects of the divorce).

If you are the Petitioner, you will have more control over how long the divorce process lasts because this depends on when you lodge certain documents with the Court. You may, for instance, be advised not to apply for a Decree Absolute (this is the final document which ends the marriage) until after the finances have been settled, especially if your spouse has a considerable pension fund.