To transgress is to rebel and to damage a relationship.  As in breaking the king's rules deliberately.
 3 Then the king's servants who were at the king's gate said to Mordecai, "Why do you transgress the king's command?" [Est 3:3 ESV]
Deliberate, intentional decision to NOT do something.

Sin is to miss the mark, not even close to the target.  
16 Among all these were 700 chosen men who were left-handed; every one could sling a stone at a hair and not miss. [Jdg 20:16 ESV]
Word translated "miss" here is elsewhere translated "sin".  This is to miss what you aimed at.  Implies an intended action, but failure to execute.

Iniquity is to be bent, twisted, as on the inside.  
13 And I declare to him that I am about to punish his house forever, for the iniquity that he knew, because his sons were blaspheming God, and he did not restrain them. [1Sa 3:13 ESV]
This is refusal to do what is right, when you know what is right.  Here is another verse:
8 And David said to God, "I have sinned greatly in that I have done this thing. But now, please take away the iniquity of your servant, for I have acted very foolishly." [1Ch 21:8 ESV]
David knew exactly what he was doing with Bathsheba, and he did it anyway, and then went all the way to murder because of it.  Bent and twisted.

Based on Hance Dilbeck's teaching on Psa 32 and 51, 2/8/17.  A Wednesday night.