Nike or Adidas? Depends what you're looking for in shoes specifically. Nike has a lot of classic shoes like the Air Force 1's and the Roshe Runs. Adidas has some awesome technology like the Pure Boost shoes. Many skaters prefer Adidas over Nike. Collectors seem to prefer Nike over Adidas. Look at these ratings for both brands to see what a bunch of other customers think.
Service wise, I've had good experiences with both companies. When it comes down to it, I honestly like them both equally. For me, it really depends on the individual shoe. Both Nike and Adidas has crappy shoes as well as awesome shoes. Why? Because we all have different preferences and tastes and because of that, it's really hard to say which company is the best. Just based on sales and awesome commercials, Nike wins. They also make Jordan shoes so that's another huge point. If you base it on who has the most comfortable shoe, I think Adidas wins with their Pure Boost shoes.
So is it style you're after? Comfort? Cost? Quality? All of that makes a difference.
Now if you are just talking about brand, then I think Nike easily wins since they're more recognizable.