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.
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.”