"Consequently" versus "consequentially"

I have to say first that consequentially is a pretty rare word. My advice would be to steer well clear of it, because nearly every time you use it, people will assume you're making a mistake and that you should have used consequently.

In principle consequentially does overlap in meaning with consequently, in that they can both mean as a result [of whatever has just been mentioned], but consequentially has more the implication of an indirect result.

Also note the usage A man of consequence, meaning an important man. In my experience, this meaning is never implied by consequent or consequently, but it often is by consequential (and presumably consequentially, but I must be honest and say I myself am not very familiar with that form at all).

It may be helpful to consider the common negated form inconsequential (normally meaning trivial, having no significant consequences, but sometimes also applied to things that do not logically follow from what's just been said).