Depending on the foundry Expert Sets could hold different configurations of glyphs. The naming implied that those fonts provided all the characters missing in the standard fonts that a typographic expert may have need of. Originally Expert Sets included only small caps, oldstyle or hanging figures, additional ligatures, often super- and subscript letters and numbers, plus some additional special characters and sometimes swash variants. Normal height capitals were absent, and their slots were occupied by other expert characters.
I am wondering if there is specific elements that characterize an expert set beyond the fact that there is additional characters provided.
There is no standard definition of "Pro" or "Expert" fonts. It was a term that started in the early 2000's when many foundries updated their fonts to include a wider character range and OpenType features. However there was no consensus between foundries on which character set or which features constituted an expert font, so currently the term means generally a font intended for professional typesetting, with enough characters / features for text, but your mileage may vary on what it includes.
Expert set fonts are additional fonts sold with professional font families. Often they would have genuine small caps as the lower-case, text figures for the numerals, swash capitals in the capital positions (in italic), and maybe ligatures at some codepoint or other (font design studios used to have standards for those; I've checked an old manual for Neutraface and you pressed shift-option 5 for an 'fi' ligature). You would switch fonts to the expert set (say from Minion to Minion-Expert) when you needed those features, and sometimes I think people would have macros to do the ligature substitution. Now that most apps (Microsoft Word excepted) can handle OpenType alternate characters you rarely see them. But there is one primitive font family like this still if you're on a Mac: Bodoni 72 comes with separate "small caps" and "oldstyle figures" fonts, and the ligatures are not substituted in, you have to copy and paste them in. Alternatively, sometimes these characters would be encoded in unusual codepoints in the font used for its normal characters, and you'd find them using a character map tool and copy and paste them in (or set up a macro to do it).