"Consequently" means "because of that" or "as a result", as in: "He had lost all his money. Consequently, he could not pay for the goods".
"Consequentially" is rarely used. It is the adverb formed from "consequential", which is an adjective meaning "resulting", as in "A hurricane blew through the town. The consequential damage was horrifying".
I suppose you could say "The consequentially occurring damage from the hurricane that blew through the town was horrifying", but it would be clumsy and even pretentious.
You will probably never have an occasion to use "consequentially".
I just had an occasion to use "consequentially", thought it might be of some assurance to those who've found themselves in a similar situation: "I could hardly tear myself from his company, and the consequentially small amount of time left in the hour..."
Sure, the word isn't essential to my context; I could have changed the wording to get rid of it, but it was the choice I automatically opted for and doesn't feel too clumsy to me.