1) Browsing time: Is a time calculation based upon dates & times from log data (Most log data does not have a browsing time field). So we have to calculate it. If there is active browsing occurring in the log data for a user from 9am til 9.34 then the browsing time would be 34mins. (depending on your browsing time setting. (I believe it is set to a 5min window by default). What this setting means is that if activity occurs within that window of time then it is part of the same browsing time calculation.
2) Duration: is usually taken from log data but is usually based on the time taken for the request for that website til when it is delivered to the pc.
So the major difference there is browsing time attempts to calculated the time spent browsing the internet and reading/looking at the website while duration just gives you the time it took to download it.
The definition of how the duration field is calculated will depend on your device/software that produced your log data.