What these cursive letters are
Both styles come from a part of Unicode called Mathematical Alphanumeric Symbols. They were added for maths, where script letters stand for particular quantities, and people borrowed them for decoration. They are characters, not a font, so they travel with the text when you paste it.
Script and bold script
Script is the thin one. Bold script is heavier and easier to read at small sizes. Neither is joined-up handwriting. Every letter stands alone, so the result looks like cursive without the letters connecting.
A few letters look different
In the thin script style, Unicode reuses older symbols for B, E, F, H, I, L, M and R, and for e, g and o, because those letters already existed elsewhere. Most devices draw them well, but a few fonts make them look slightly unlike their neighbours. Bold script has no such exceptions.
Where it works and where it does not
Most chat apps, social profiles and text boxes accept it. Some usernames, email addresses, file names and passwords do not. People searching for your name will not find cursive text as ordinary words either, so do not use it for anything you want found.