This led me to look up what these two words have to do with each other. From Wikipedia:
The word piety comes from the Latin word pietas, the noun form of the adjective pius (which means “devout” or “good”). Pietas in traditional Latin usage expressed a complex, highly valued Roman virtue; a man with pietas respected his responsibilities to gods, country, parents, and kin.
Morality (from the Latin moralitas “manner, character, proper behavior”) is the differentiation of intentions, decisions, and actions between those that are “good” (or right) and those that are “bad” (or wrong).
So, moral behavior would be derived from the piety one felt and what that respect would mean. Morality, to put it bluntly, cannot be objectively quantified. All morality is relative.
(We usually say “filial piety” when we mean respect to the family to distinguish it from respect for one’s god. It is interesting that the original Latin word could be used to talk about piety towards one’s country. Today, we use the word “patriotism” for that.)
As a result of this etymological search I don’t see a separation between piety (in the more general, Latin, sense) and morality. It looks to me like morality flows from the pieties one maintains.
That is, if you don’t have piety for my God, you won’t be motivated to do things that are in the moral system I have that results from the pieties I maintain.
Most of us have piety (that is, respect) for at least some sense of order and life, so prohibitions against, say, murder fit into both of our moral frameworks.
However, I have piety (respect) for God that others don’t share. Many people have a great piety towards science or reason that I don’t share. So they will do things that I wouldn’t do and I’ll do things that they won’t do. (For example, I am more ready to embrace the irrational, and many people don’t bother with loving God.)
So, when you hear people ranting about how immoral some people are, they’re absolutely right. However, they can’t expect anyone who doesn’t share their pieties to share their morality.