When math alphabets were added to Unicode (Uniocde 3.1) "Calligraphic" and "script" were considered to be simply font variants with no semantic distinction and and unified into the alphabet starting U+1D49C MATHEMATICAL SCRIPT CAPITAL A
This wasn't universally popular as some fonts use a more "calligraphic" or "Chancery" style and some use a more cursive "script" style. Some fonts have both accessible via font feature settings but not all applications can access those and its hard to use them while still allowing font variation as different fonts specify different stylistic variations.
Unicode is (off and on) wondering about addressing that and having two (or three) math alphabets so as to make these more easy to be used to give a semantic distinction, see Murray's post here
If people do have examples of published (preferably peer reviewed) documents that make use of these two styles in semantically distinct way, that would be useful data to add to the discussions on whether to add a new alphabet to Unicode.
But the current official Unicode line would be (as far as I understand it) that it's like choosing between Times and Computer modern, they are very visually distinct, but the choice is just personal preference with no semantic implication.