Similarities between connectives 就 and 而? 而 - is a connective like "and" in some situations. Etymologically this meaning comes from how hair in someone's beard are close together.
就 - is a connective in other situations. Etymology information is lost to us. 

Can we say that sometimes they are equal in their meaning and function? In what circumstances are they equal?
Actually, these two words are very different, and it's the difference - not the similarity - you need to grasp. 

Both of these words indicate a rhetorical "following from," but 就 means "to follow from" in the broadest, simplest sense, rather like an English "then": 

When it's used as a 'verb', it's intended to show that one thing follows as a consequence of another. An excellent example of how this usage can be precisely mapped in English is below:

"若這樣就那樣“ - "If this, then that."
"他吃了整之火雞，那他就飽了。“ - "He ate the whole turkey, so he, then, [is] full!"

As you can see, "就“ as a "verb" is easily understandable as an English "then", as well. 

Now consider this:

"我渴死了“  - "I'm thirsty!"
”那，你就去喝水吧!“ - "So go drink, then...."

The many instances this phrase can be used - whether humorously, as a reprimand, as an exclamation, as an acquiescence, as mockery - all pretty much depend on "就" being the simplest, most common-sense idea anyone could have. 若一就二 - like that.

“而” is more like an "and" than a "then" - however, it is more specific and narrow because it further indicates transcendence, or moving beyond the expectations of discourse. In that sense, it's much more like "moreover" - so whether one uses it as "and," "also," "but," "however," or any other sense, it always has that meaning of saying "moreover" behind it - 

"我吃飽了，而吃太飽雅。。。“ -
"I ate, moreover {but/also/however}, i ate too much."

男： “昨晚我們去看弟弟演一出布萊希特戲..."
女： "而他演的好好看啦！“

Guy:  "Last night we went to see my little brother act in one of Brecht's plays."
Gal:  "Moreover {And/But/Also} he really acted so well in it!"

In that exchange above, "而“　can be used to indicate a "but" - 

Guy: "We went to see my brother act [implied: and i expected him to be awful]...."
Gal:  "But he really looked great up there!"

Or an "and" that draws attention to the importance of what comes after, in contrast to what came before: 

Guy: "We saw my brother..."
Gal: "And it was amazing!"

Generally, in modern Chinese it's going to indicate a "moreover" of some sort - but depending on the rhetorical context, it can be translated into other languages in any of many ways.