These two words have been haunting me for quite long. Please provide some logic to understand and use them efficiently.
Ex: Regrettably/Regretfully, she could not see her father for the last time.
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Garner's Modern American Usage (p705) has an entry on the two words:
Errors made are regrettable; the people who made them should be regretful. The most common error is to misuse regretful for regrettable, particularly in the adverbial forms.
So you need regrettably in your example:
Regrettably, she could not see her father for the last time.
Regretfully is less likely to start a sentence. A possible example is:
Regretfully she admitted to having been unkind to her father the last time she saw him.
Shoe is right that the adjectives regretful and regrettable cannot be used interchangeably:
The adjectives regretful and regrettable are distinct in meaning: regretful means ‘feeling or showing regret’, as in she shook her head with a regretful smile, while regrettable means ‘giving rise to regret; undesirable’, as in the loss of jobs is regrettable.
However, it is not true that the adverbs regrettably and regretfully have the same distinction, at least in non-formal contexts:
The adverbs regretfully and regrettably have not, however, preserved the same distinction. Regretfully is used as a normal adverb to mean ‘in a regretful manner’ (he sighed regretfully), but it is also used as a sentence adverb meaning ‘it is regrettable that’ (regretfully, mounting costs forced the branch to close). In this latter use it is synonymous with regrettably. This is disliked by traditionalists and should be avoided in formal contexts.
Both words have the same root: regret, meaning to feel sadness, repentance, or disappointment, but they each have their own way of dealing with it.
Regrettably describes something that deserves regret, and is used like the word "unfortunately." Regrettably is like bad luck, and it often kicks off a sentence:
"Regrettably" is used for a situation that is unfortunately not as you would like it to be. It doesn't assign any fault, and doesn't say you could have done anything differently.
"Regretfully" is used if there is a regrettable situation, you caused it, and you are sorry that you caused it.
"Regrettable" describes a situation, "regretfully" describes the feelings of the person who caused it.