An adverbial phrase (see this Grammar Monster page) is simply a phrase that functions as an adverb. In the example
... she said with a professional smile.
"with a professional smile." is an adverbial phrase: it tells the reader how she said it. ("professional" in this case is an adjective, modifying "smile".) In the example
... she said, smiling professionally.
The word "professionally" is an adverb, modifying "smiling". It tells the reader how she smiled, which is something she was doing as she spoke.
The actual meaning in these two cases is much the same, even though the grammatical structure is different. The difference here is merely one of style. In some cases this will change the focus or emphasis, or even the meaning, at least in a nuance. Here there is really no difference in emphasis, let alone meaning. It is often possible to write things in different ways, sometimes with no difference in meaning, sometimes with a subtle but significant difference.
By the way, neither of the above examples is a complete sentence, which is why I have given them starting with an ellipsis and a lower-case letter.
The other two examples from the question are a bit different.
- He looked up from from his book and, in a monotone, said, "Congratulations."
- He looked up from from his book and, monotonously, said, "Congratulations."
In the first of these, "in a monotone" is an adverbial phrase. It tells the reader how he said it. In the second, "monotonously" is an adverb. But here there is a difference of meaning, because ""monotonously" does not just mean "in a monotone". As the Cambridge deinition puts it, the first sense is "in a way that does not change and is therefore boring". The Free Dictionary gives as sense 2: "Tediously repetitious or lacking in variety. See Synonyms at boring."
The first example merely describes the actual tone of his speech, while the second strongly implies that it was boring to listen to. Here the change of grammatical form comes with a change of meaning. If the intent was only to describe the physical tone, then the second example would be ill-chosen.
English offers many near synonyms and related words. Sometimes the choice matters. As Mark Twain wrote, "It is the difference between the lighting and the lighting-bug".