This is what I mean, do you usually prefer:
Hydro Pump or Surf?
Thunder or Thunderbolt?
Fire Blast or Flamethrower?
Blizzard or Ice Beam?
I personally go on Accuracy, it pisses me off when one of my moves miss, I think "if I had taught that other move it would have hit".
Well there's also PP for me to factor in as well, at least when it comes to in-game battling, I don't really want to use Hydro Pump as it's only 5, whereas Surf is 15, giving me more use out of the move without having to go back to a Pokemon center or use items that I never use(Elixirs and the like, also don't really use PP Ups so I don't really bother increasing the PP, although 8 isn't that big of a difference to me anyways, especially when Surf would end up with 24).
In general, I'd go with the more accurate moves that have a decent enough PP, at least when it comes to in-game battling. I take more chances for Battle Facilities type things, PP isn't as big of deal either in those cases as it restores after every battle. I still would usually prefer the more accurate moves though, however, I'd use Thunder and Blizzard over Thunderbolt and Ice Beam if I would be using Rain or Hail for the battle.
Also, there are cases where I'd go with less accurate moves due to a lack of another option, such as with Focus Blast where there isn't much other options for another Special Fighting type attack for most Pokemon.
But in general, I go with more Accuracy over more Power.