Apparently they have no difference whatsoever apart from verscheidene being more formal
http://webcache.googleusercontent.com/search?q=cache:n-5iUYOHBcoJ:www.dutchgrammar.com/en/%3Fn%3DPronouns.Id13+cd=2hl=enct=clnkgl=itclient=opera [link sometimes is badly displayed, in which case copy and paste it]
I notice that the moderator has still not commented as to why 'The woman has a variety of apples' is not accepted. Please do comment. 'Various apples' in English can include apples that are just slightly different from each other, perhaps slightly different in size for example but still of the same kind, whereas 'variety of apples' implies that the difference extends to different types of apples, and this I think is a more faithful translation of 'verschillende'.