utf8_general_ci and uft8_unicode_ci have nothing to do with "storage", only with "collation" (that is sorting, as in ORDER BY, GROUP BY, WHERE a < b, etc)
uft8_unicode_ci is arguably the 'better' collation, but slightly slower.
You apparently have the "character set" set to utf8.
IF you want to change the collation, do
ALTER TABLE foo MODIFY COLUMN bar ... COLLATION uft8_unicode_ci.
This does NOT change the 'storage', only the actions performed by ORDER BY, etc. Well, it does have to change any indexes that depend on `bar`; but the actual data is not changed. See
http://dev.mysql.com/doc/refman/5.0/en/alter-table.html
IF you need to change the data, that's a different topic.