"Nobody goes to bed in Madrid until they have killed the night." - Ernest Hemingway
Although I have never lived in Madrid, I have spent a few summers on the outskirts and when I lived further afield (Extremadura) I would still have to pass through the capital on my way home.
As such, I have visited Madrid more times than I can count. It may not be my favorite Spanish city but it is the thriving centre of the country and great for a weekend away.
There is something very authentic and quaint about the winding streets of Madrid, many parts feel almost provincial. This bustling city feels lived in and well loved.
Of course the Gran Via is more reminiscent of New York and its other large city counterparts. Great for shopping but certainly very touristy.
My favorite thing to do in Madrid is to take the teleferico out to Casa del Campo. The views are breathtaking and it is a good relief from the intense summer heat. Taking you out to the spanish countryside with stunning views of both rural landscape and city, this cable car is also seriously interesting and educational. With stunning views of the Almudena Cathedral and Royal Palace, a voice over the speakerphone gives you historical and cultural information throughout the ride. For roughly 5/6 euros return ticket, it's not only a different way to view Madrid, its also great value!