Is severe but short-term pain worse than moderate but prolonged pain for the body?
The biological part of the answer to your question depends on the thing causing the pain. Burning for example, a large burn can be very life threatening biologically but small burns are not as harmful comparatively. But the psychological harm they would cause is phenomenal. This is because large burns open up the body to infections, dehydration and a lot of other harm. 

Let's take stabbing as another cause of pain. One stab in a less dangerous place missing all important structures may be not that harmful but may be painful. However multiple small stabs into important structures like arteries or spinal cord could be horribly destructive. 

Then let's look at headaches or nerve pain. Here it is only the pain and usually there isn't any destructive or harmful process. Most individuals would take one really, really bad headache followed by never having a headache then chronic daily "tension" headaches. 
Psychologically though usually the chronic daily pain is the worst. Particularly as without adequate pain management from the start, it is less responsive to painkillers (analgesia). It frequently adds to depression if it does not cause it itself and can lead to suicide. Also painkillers themselves may cause chronic pain which does not help. 

In summary, acute severe pain suggests something is seriously wrong and needs to be dealt with immediately biologically as there is likely a damaging process underlying it. However if this event is survived, it may be linked with high disability from any biological process that underpinned the pain. Chronic daily pain suggests a long continuing process. The cause of this may be life threatening, such as cancers, but is usually less dangerous at least compared to acute severe pain biologically. However the psychological aspects of this pain are seriously important as they can be particularly detrimental.