You can check and compare sort orders provided by these two collations here:
utf8_general_ci is a very simple collation. What it does - it just
- removes all accents
- then converts to upper case
and uses the code of this sort of "base letter" result letter to compare.
For example, these Latin letters: ÀÁÅåāă (and all other Latin letters "a"
with any accents and in any cases) are all compared as equal to "A".
utf8_unicode_ci uses the default Unicode collation element table (DUCET).
The main differences are:
1. utf8_unicode_ci supports so called expansions and ligatures, for example:
German letter ß (U+00DF LETTER SHARP S) is sorted near "ss"
Letter Œ (U+0152 LATIN CAPITAL LIGATURE OE) is sorted near "OE".
utf8_general_ci does not support expansions/ligatures, it sorts
all these letters as single characters, and sometimes in a wrong order.
2. utf8_unicode_ci is *generally* more accurate for all scripts.
For example, on Cyrillic block:
utf8_unicode_ci is fine for all these languages:
Russian, Bulgarian, Belarusian, Macedonian, Serbian, and Ukrainian.
While utf8_general_ci is fine only for Russian and Bulgarian subset of Cyrillic.
Extra letters used in Belarusian, Macedonian, Serbian, and Ukrainian
are sorted not well.
The disadvantage of utf8_unicode_ci is that it is a little bit
slower than utf8_general_ci.
So when you need better sorting order - use utf8_unicode_ci,
and when you utterly interested in performance - use utf8_general_ci.