Azimah is the original binding legal ruling; rukhsah is a specific legal relief granted because an accountable person faces a recognized excuse.
The first kind grants an exceptional allowance while the prohibition remains, so steadfast nonperformance may be rewarded. The second makes an act permissible during genuine necessity; refusing a necessary life-saving measure may then be sinful.
For each case identify three things: What is the original ruling? What kind of excuse has arisen? Does the concession leave the prohibition in place or make the act permissible for the emergency? The practical conclusion follows from those distinctions.