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.
utf8_unicode_ci 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.

(c) Alexander Barkov

You can check and compare sort orders provided by these two collations here:
http://www.collation-charts.org/mysql60/mysql604.utf8_general_ci.european.html
http://www.collation-charts.org/mysql60/mysql604.utf8_unicode_ci.european.html