among them; especially; above all
Refers back to a group or range of things and singles out something as especially notable. Similar to 中でも, and often a little written or rhetorical in tone.
See also: 中でも (なかでも)
Japan has many famous festivals, but among them the Gion Festival is especially well known.
at the same time; amid; with both qualities
Often used in patterns like Xの中にもYがある or Yが感じられる, showing that another feeling or quality exists within the same situation, sometimes in contrast. It is not a general-purpose replacement for 同時に.
In his expression, there was sadness even amid his happiness.
同時に simply means 'at the same time', while 中にも in this sense suggests one feeling or quality is present within another situation, often contrastively.
特に is a general word for 'especially' usable anywhere, whereas 中にも in this sense specifically points back to a stated group or range.
A transparent phrase built from 中, “inside; among,” plus the particles に and も, giving senses such as “among even these” or “also within.”