You will need a function in Time to return the minutes. Here's a revised Time.
class Time
{
   public:
      Time();
      Time(int hour, int minute);
      int time_GetHour() const;
      int time_GetMinute() const;
   private:
      int hour;
      int minute;
};
int Time::time_GetHour() const
{
   return hour;
}
int Time::time_GetMinute() const
{
   return minute;
}
Your definition of DaTime is a little bit unclear. Do you need to store start and end as well as duration? Can you get away with storing either start and end or duration? What purpose does the input Day day serve in construction of DaTime?
If you have already computed duration, DaTime::dt_Difference can be altered a little bit to use duration instead of relying on the member functions of Time.
int DaTime::dt_Difference(DaTime &a, DaTime &b) const
{
   int diff = 0;
   if(dt_LessThanOrEqual(a,b))
   {
       diff = b.duration - a.duration;
   }
   return diff;
}
However, the way you have decalredDaTime::dt_Difference, it will be awkward to use. You have to have lines like:
DaTime dt1;
DaTime dt2;
DaTime dt3;
int difference = dt1.dt_Difference(dt2, dt3);
Notice that the state of dt1 has no impact on the return value of dt_Difference. Is this how intended to use it? A better alternative would be to change the declaration of dt_Difference to be:
int dt_Difference(const DaTime &a) const;
Then, the implementation would be altered to be:
int DaTime::dt_Difference(const DaTime &a) const
{
   int diff = 0;
   if(dt_LessThanOrEqual(*this,a))
   {
       diff = b.duration - this->duration;
   }
   return diff;
}
Then, your usage could be changed to:
DaTime dt1;
DaTime dt2;
int difference = dt1.dt_Difference(dt2);
In addition, there is no need to have names time_GetHour and time_GetMinute. time is redundant since you are dealing with the class Time. You could change them to getHour and getSecond. Similarly, you could also change dt_Difference to operator-. You could also replace the use of if(dt_LessThanOrEqual(*this, a)) by if ( *this <= a ). The second form is more readable than the first form.
In summary, your code could be changed to look like the following:
class Time
{
   public:
      Time();
      Time(int hour, int minute);
      int getHour() const;
      int getMinute() const;
   private:
      int hour;
      int minute;
};
int Time::getHour() const
{
   return hour;
}
int Time::getMinute() const
{
   return minute;
}
class DaTime
{
   public:
      DaTime();
      DaTime(Day day, Time start, Time end);
      int operator-(const DaTime &a) const;
      bool operator<=(const DaTime& rhs) const;
   private:
      int duration;
};
int DaTime::operator-(const DaTime &a) const
{
   int diff = 0;
   if( a <= *this )
   {
       diff = this->duration - a.duration;
   }
   return diff;
}
DaTime dt1;
DaTime dt2;
int difference = dt2 - dt1;