不名誉 means 'dishonorable' or 'disgraceful' and focuses on loss of reputation, whereas 破廉恥 emphasizes the lack of moral shame.
Etymology
Compound of 破 (ha, 'break, destroy') and 廉恥 (renchi, 'sense of honor, shame'). The exact historical derivation is uncertain, but the combination conveys a complete loss of moral decency.