Now I was also told that Half equivalence point is when [HA] = [A−] , the concentration of a weak acid = concentration of conjugate base.
That is correct. You make the weak acid in situ when you titrate a weak base with a strong acid, or when you titrate a weak acid with a strong base. At the half equivalence point, the pH is roughly equal to the pKa of the weak acid.
What is the difference between a half equivalence and an equivalence point?
The equivalence point is what it always is. The point when you added stoichiometric amounts of titrant to the analyte. In the weak acid / strong base case, it means that you have quantitatively turned the weak acid into weak base, and don't have any excess of strong base (all was neutralized). The pH would be that of the weak base. In the weak base / strong acid case, it means that you have quantitatively turned the weak base into weak acid, and don't have any excess of strong acid (all was neutralized). The pH would be that of the weak acid.
Based on what i have been told, they sound the same to me.
Compared to the equivalence point, you have to add half the titrant to reach the half equivalence point, so they are different.