TL;DR The Unicode consortium recommends using the latin letter where possible and not the numeral, which where included for compatibility with East-Asian typography.
The full story : (with justification of the above assertion)
Unless you are doing some East-Asian typography, using the (non-archaic) Roman numeral characters from unicode (U+2160 — U+217F) is a hack.
These character have been included for compatibility with pre-Unicode East-Asian standards. These characters stays vertical where the East-Asian text is typeset from top to bottom, while usually, text in Latin characters (e.g. names) are written sideways in this context.
To quote the last version of the Unicode standard (v 7.0, chap. 22, p. 20):
Roman Numerals. For most purposes, it is preferable to compose the Roman numerals from sequences of the appropriate Latin letters. However, the uppercase and lowercase variants of the Roman numerals through 12, plus L, C, D, and M, have been encoded in the Number Forms block (U+2150..U+218F) for compatibility with East Asian standards. Unlike sequences of Latin letters, these symbols remain upright in vertical layout. Additionally, in certain locales, compact date formats use Roman numerals for the month, but may expect the use of a single character.
So, in theory, the distinction between Roman Numerals and letter is a matter of rich text, like italics, a font change, or optional ligatures. That said, as @Wrzlprmft shows, some font use it to avoid a font change for each Roman numeral while keeping a good typography.
The existence of a character for XII and not for XIII implies that there are several different encodings the same numeral, which leads to difficulties in text search : If you write about Louis XII and Louis XIII, you will probably write XIII as X+I+I+I, but will you write XII as a single character ? Or as X+I+I to have a consistent display with XIII ? There is no single good answer to this question while using the Roman Numeral Characters, and that’s why the Unicode consortium recommends using the Latin letters when possible and not the numerals.
Edit : added the TL;DR assertion in the beginning