adverbsdifferences
I saw this on eBay's website:
You can sell multiple items that, even though they are against eBay
policy, don’t get you caught. Never the less, they are against the
rules and can result in you losing your eBay account.
Is never the less the formal version of nevertheless?
Personally I would put more of a regular scheme to frequently, as the word comes from the same stem as frequency (a sine curve of certain wave length, therefore repeating itself in a certain pattern). Often would denote more of an irregularity.
Maybe this is a déformation professionel, as I am programmer with a math background, or I derive it from German and Latin languages, for example French:
fréquenter quelquechose would mean to regularly go somewhere.
Therefore I would hear this
Effectually is effectively archaic. Don't use it.
Note that OP's definitions evidence the fact that the two words are synonyms in the adverbial sense of "with efficacy", which normally implies purposefully and efficiently.
On the other hand, my own usage at the start of this answer has more the sense of "in effect, to all intents and purposes, in practice" (a meaning which also originally attached to effectually). But since most people probably won't be so familiar with the archaic form, presented with...
High dosage of morphine effectually/effectively ended his life.
...they might well tend to interpret effectually there as meaning the morphine was administered in order to give him a painless death, where effectively would often be seen as implying that death was more of an unfortunate/unforeseen/accidental consequence of the pain-killing treatment.
For all that, I would not recommend using effectually in any non-facetious circumstances.
Best Answer
Nevertheless is a conjunctive adverb:
Never the less is probably a typo. When it appears as three separate words, it means something quite different. One valid way of using this three-word phrase would be:
A quick Google NGram search shows that never the less is quite rare.