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.'
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 mentions that a student refused to behave. B asked for to know specifically which student refused to behave.)