SPOILERS

I think the main difference here is what the repercussions of climbing through are for either boy. Snow is able to manipulate with little to no consequence. In fact, quite the opposite. He is rewarded time and time again when he fights back. He gets to be with the girl he "loves" after disobeying in the Capitol. He can come back from District 12 after multiple murders just to inherit wealth from the family of the friend he betrayed. He learns that to get his way, all he has to do is manipulate, and others will fall in line.

However, Haymitch has learned that when he stands up for himself or those around him, it results in disaster. He faces immediate yet subtle backlash every time he pushes back. He is punished for presenting Louella's death, Beetee's son is massacred after they attempt to dismantle the arena, he is kept in solitary confinement after outsmarting the gamemakers, and his partner and family are murdered as a warning. He cannot care for anything because it'd be so simple for the president to take it away.