As implemented in NWN, there essentially are no differences between the two classes as far as RP/lore considerations go. One or the other may work better in a particular build for reasons of game mechanics, but aside from that it is basically six of one, half a dozen of the other. In PnP, as Malagant also notes, the big difference between the two is that while Paladins are limited to serving deities and causes compatible with a lawful good alignment, Divine Champions may be of any alignment and serve any deity. This difference is still somewhat true in NWN, as COTs can be any non-evil alignment (less restrictive than LG only), though excluding evil makes it a much less important difference, and it becomes less significant still if one interprets COTs as literally champions of Torm (a LG deity) rather than as generic divine champions.
I suspect Bioware may have implemented them as they did because of the Divine Champion's Smite Infidel ability (like Smite Good but usable against anyone with a different patron deity). They may have thought that this would not have worked well in a typical NWN context, or just did not want to take the trouble to design a brand new ability. This perhaps led to them being given Smite Good instead, and that in turn made an evil alignment for them not make sense.
