Here is a simple difference explained by definition:
Average session duration per session = total duration of all sessions(sec)/number of sessions
Average session duration per user = Total duration of all sessions(sec)/number of unique users
For example, the total duration of all sessions is 25000s and the number of sessions is 125
Average session duration per session = 25000/125 = 200s
And if the number of unique visitors is 250, average session duration per user would be 25000/250 = 100s.
You will see both the metrics in the behavior section while using the advanced segments in google analytics.