Is there a difference between [shinkeishitsu na] 神経質な and [fuan na] 不安な?

Question:

Are there any specific times when I should use one over the other?

Answer:

神経質＝noun usages:nervousness.

神経質な
nervous.

神経質に
nervously.

神経質な人
a nervous person; a person with a nervous temperament.

彼はひどく神経質だ
He's very nervous./話 He's all nerves.

彼の手は神経質そうに震えていた
His hand was skaking nervously.

The usages of 不安　is as follow:
1 〔心配〕uneasiness ((about)); anxiety ((about)); worry ((about, over))

不安 (そう) な顔つき
「a worried [an uneasy] look

不安な一夜を過ごした
I passed an anxious night.

不安な思いで家からの手紙を待った
I waited for the letter from home with anxiety.

父の安否が分からず不安です
I am worried about my father's safety.

いつも地震に対する不安におののいている
He is in constant fear of earthquakes.

突然得体の知れない不安に襲われた
Suddenly I was seized with an inexplicable disquiet.

うそがばれはしないかと不安だった
He feared [was afraid] that his lie might be detected.

2 〔不安定〕insecurity; 〔社会的動揺〕unrest

政界［社会］の不安が株の大暴落を起こした
Political [Social] unrest caused a drastic fall in stock prices.

不安な立ち場にいる
be in an insecure position

不安神経症｜generalized anxiety disorder; anxiety neurosis