I think it's the same as the difference between 'may' and 'can' in english. The problem is that there's no past tense form of 'may' in english so we end up using 'were allowed to', whereas the dutch just use mochten.
In English, you could use "may" for the present tense or "might" for the past. I don't know whether "might" would be accepted here. It does sound a bit old-fashioned and stilted to me. However, the sentence is artificial anyway.
I think that whereas "can" , in English, can mean both being able to do something or having permission to do something, the same doesn't hold true for the Dutch word "mochten". I understand it the way iGeo85 explains it.