You could artifically introduce the year for your comparison. E.g.
string s1 = "15/03";
string s2 = "01/03";
DateTime d1 = DateTime.Parse(s1 + "/2014");
DateTime d2 = DateTime.Parse(s2 + "/2014");
TimeSpan t = d1.Subtract(d2);
You would need to ensure that s1 is always the later "date" and you may want to consider the overlap over the end of the year e.g. 20 December to 03 January. But the principle would still work - just choose an arbitrary year.
Watch out for leap years too