I think that Azula broke down more because all her life she's been brought up to essentially fight by her father's side and be a fighting machine. And then she realized that her father didn't care about her anymore. He just gave her an insignificant role of Fire Lord and didn't even bring her along to destroy the earth kingdom. The only person that she *thought* loved her a lot brushing her off plus her bffs betraying her plus her mom thinking she was a monster equals crudloads of emotional damage, i think.