Difference between adverb and adverbial adjunct

Solution 1:

Today functions as an adverb of time, telling when he arrived. Dictionaries usually present today as an adverb. That's the most straightforward analysis.

If you wanted, you can call an adverb an adverbial, and describe it as an adjunct that provides additional information to the sentence. Oxford Dictionaries explains the respective terms in ways that allow this analysis. (Adverbs are also considered adverbials; as adjuncts they provide extra information,) The result is that it treats a very similar word (yesterday) in a similar sentence (She visited ... yesterday) as an adverbial adjunct:

Adverbial adjuncts can provide extra information about: (...)

when things happen: (...)

• I can’t sleep at night.

• She visited her family yesterday.

So yes, you can call today, like yesterday in the previous example, an adverbial adjunct.

There are also other explanations for what's going on here. Some guides like The Cambridge Grammar of the English Language (CGEL) analyze today, yesterday, and related words as pronouns (link is to StackExchange answer on subject), and would describe this as a noun phrase used as an adjunct of temporal location. The result is similar to calling it an "adverbial adjunct," but CGEL goes to such lengths to better account for cases where today is clearly not functioning as an adverb.