Yes, there are several basic differences in the usage of "Dankbarkeit" and "Erkenntlichkeit".
As noun the word "Erkenntlichkeit" is effectively not used any more in the today's language. Even if the Ngram Viewer indicates some extremely light usage of the word "Erkenntlichkeit" in the 21th century, zooming into the hits reveals that all of them are simply quotations of older works.
"sich erkenntlich zeigen" is the only phrase with "erkenntlich" (meant as "thankful" not as "perceptible") known to me which is still in use today. But usually this phrase doesn't mean "to be thankful" but rather "to do something or to serve in return for a service", often used as an euphemism for services or payments in return in deals which are not documented by the parties as of their illegal or immoral nature.
So, for today we are left alone with "Dankbarkeit" in all its forms:
Ich verspüre Dankbarkeit.
Ich danke Ihnen für Ihre Aufmerksamkeit.
Für eine schnelle Bearbeitung wäre ich Ihnen sehr dankbar.
"gratitude for another's action" can also be expressed by "verbunden" / "Verbundenheit":
Für eine schnelle Bearbeitung wäre ich Ihnen sehr verbunden.
Für alles, was Du getan hast, fühle ich mich Dir sehr verbunden.
The usage of "verbunden" has a refined connotation with a touch of emotionality.