Imagine the series was written to be about wizards and witches, but they didn't use wands and instead simply cast spells from their hands/fingers.

Would the series and its plot be any different and if yes, how?

The plot with Harry's and Voldemort's wands being connected would clearly be gone, that's for sure.

Also, how would this affect fanfiction?

 

Well, Harry might actually join Voldemort in this universe, because the disarming charm is clearly a dark art xD
Detaching limbs aside, I'm curious how the thing with disarming making the loser's magic "subservient" to the winner would work out throughout wizarding society.

The part about connected wands could be readily replaced with them being connected through souls/blood. Not sure what the equivalent of the Elder Wand would be though.
What's interesting is how that would impact wizard's sense of superiority, or gatekeeping magic, like goblins being prohibited from owning wands, what would be the ramifications of their non-existence?

Fanon-wise, a lot more ambi-dextrous wild-west finger-slinging spell-shooting super-Harry stories, that's for sure. Probably Gryffindor's sword would be made out to be some kind of magic amplifier. Though I'm sure there are plenty of stories like that already.

 

I think if there were no wands we may see more applications for enchanted objects like Gryffindor's sword or for potions. If wandless magic is slow/more difficult still, these advanced preparations may be more important for the world.

If wizards can pull of an Alohomora or any spell with their fingers as easily as with their wands, then we might see more themes about life debts and connections between wizards [edit: I'm thinking of overall themes since the "wand chooses the wizard" in Books 1 and 7 would not be as important]. Harry's clemency ends up saving his life when he is imprisoned by Pettigrew and the Malfoys, so that theme would remain, I imagine. Snape would still be in a kind of debt to Harry, perhaps as much because James saved his life and because he feels guilt for Lily's death. Dumbledore conquers Grindelwald because he had saved him from Aberforth's rage during their fight and G. was present for the manslaughter of Ariana. So basically more "deep magic."