Do we use the -e at the end of the adjective because it's "de vrouw"? What if it were a het noun... such as "een moeilijke paard"?
I think it means "rich in importance" - so just important, not rich money-wise.
A woman generally describes a person who is biologically female (or declares themself female) and either pubescent or over the age of 18, whereas a lady is a specific type of woman - namely a noble, classy etc. one (seen from a historic point of view). You can also address a woman with "lady" if you want to show your respect (e.g. "Mylady ...")