Particularly doesn't necessarily refer to a finer detail, but rather a section which may exhibit a greater factor. It is from the word 'particular.'
Specifically can be used provide or express greater detail of an exclusive situation. It is from the word 'specific.'
Example:
She often pronounces words wrong, particularly 'library.'
(She pronounces words wrong, but library she pronounces wrong more often or worse than others)
A: Hey, don't erase the chalk board! That's my job!
B: The teacher specifically asked me to do it.
(The teacher asked B specifically to do it. This is meant to express that B was asked by the teacher to do it, and that the job is exclusive to him)
A: One of the students refused to behave.
B: Which one, specifically?
A: John.
(A mentions that a student refused to behave. B asked for to know specifically which student refused to behave.)