What's the difference between 应急出口 (yìngjí chūkǒu) and 安全出口 (ānquán chūkǒu) which both mean "emergency exit"?
They are just two different ways to tell you where you can go safely in emergency case.
As you said,

应急 = emergency
(actually emergency is 紧急; 应急 is &quot;for emergency&quot;, which is short for 因应紧急)
安全 = safety

And they really are

应急出口
(safety) exit for emergency
安全出口
safety exit (for emergency)