i agree with 潇湘草 烦躁is used to describe someone's mental state and 急躁 is for somebody是 nature so you can feel 烦躁 but you can't feel 急躁 and 烦躁 emphasize being annoyed while 急躁 focus on being in a hurry

烦躁指心情 急躁指性格

烦躁：dysphoria 心中烦闷不安，急躁易怒，甚则手足动作及行为举止躁动不宁的表现。(The heart is restless,irritable,even the hand movements and behaviors have restless performance.) 急躁：quick/impatient/ 1.He was so impatient that I could hardly hold him back. 他是那样急躁，我简直拉不住他。 2.He is quick in temper. 他生性急躁。 3.Impetuosity will only make things worse. 急躁只能坏事。 4.His temper is of the quickest. 他的脾气是最急躁的。 5.She fussed about, unable to hide her impatience. 她激动不安，无法掩饰她的急躁。

急躁中的急，更偏向是有一定原因引起的，而烦躁就不会，可以说自己因为无聊而烦躁，但不能说因为无聊而急躁。区别不是十分明显。