Does it matter who files for divorce first?

In a word, yes

The plaintiff (the party filing the case) works on his or her own timeline, at his or her own leisure. The defendant (the party responding to plaintiff's filing) has strict deadlines to follow. In some situations, defendants have as few as 14 days after getting served to hire an attorney ... and get objections to ex parte orders drafted and filed. There're tactical advantages to filing first. However, don't despair if you get served with papers -- I represent defendants in around half of my cases.