The aerodynamics are complicated but lets not simplify it so much as to get the explanations wrong.
Winglets do increase the aspect ratio by increasing the effective span while not increasing the physical span of the wing. They have the same effect as raked wing tips which aren't canted and are what Boeing has opted for in the 747-8. Ideally aircraft would have very long thin wings but this causes nightmares for the structural engineers and makes for bigger hangers/ground handling problems etc.
By increasing the span you reduced the spanwise flow of air which flows from the bottom surface to the top around the wingtips. This flow of air causes tip vortices which while pretty, increase the induced drag. Winglets reduced that flow by increasing the aspect ratio, reducing the size/strength of the tip vortices and thus reducing the induced drag.
They don't capture lift or produce lift in the vertical sense themselves, however due to the decreased downwash behind the wing there is small increase in lift produced by the wing - which in reality then increases the lift induced drag again!. They also increase the parasitic drag and they do produce a small amount of apparent thrust. These factors along with the structural/weight implications offsets some the gains and makes it a bit of a balancing act to give the best improvements.
And with that in mind..back to the original question: Winglets/Wingtip fences/raked wingtips while structurally different are all various solutions to the same problem of wing tip vortices and induced drag.