Hi Elena. When used adjectivally, “deep” tends to be used more physically/literally (a deep valley) and “profound” more metaphorically/figuratively (a profound impact). As adverbs, both words are typically more figurative in their use and are (near) synonyms. However, “deeply” sounds a bit more direct/straightforward while “profoundly” sounds more literary in style or more emphatic. I guess one could say that “deeply” is a bit less formal, but I wouldn’t call it “informal” strictly speaking (they both seem somewhat formal to me).

Basically deeply means strong feeling. For example: He loved her deeply. She was deeply hurt by the break up. While profoundly means deep or a significant impact. For examlle: The experience profoundly changed her. His death profoundly affected the family. Sometimes they can be used in a similar way but “profoundly” sounds more significant or life-changing, while “deeply” is more natural for emotions. I hope this helps

That's a difficult question! The Cambridge Dictionary lists them both as synonyms, but I think you're right that "deeply" may be more informal. It also has a connotation of physical depth, as in "deep dive". I might say "I was profoundly bored by the lecture", for emphasis and slightly tongue in cheek, but "I was profoundly offended" sounds unnatural to me. Looking at the examples in the Cambridge dictionary, you can only really use "deeply" in "she sighed deeply" and "breathe in deeply". Some of the other examples might be more impactful without the word "deeply" - "he was [deeply] wounded by her fierce criticism." That's just my take on it - I'm not a language teacher!