Every time I go into settings and test my connection, it comes out as connected. I can use the internet browser just fine, but whenever I try to use any apps like netflix, it says there is no internet connection. Please help!! Thanks.

I have the same problem sometimes. Browser works fine but Miiverse or eShop won't connect.

I need to use a wired connection. Wireless worked on launch day, and never again after that. It has no problem passing the connection test, but once I go to actually use it, it says there's no connection.

I had this same problem.
There are plenty of threads on this. Its best to read this Kotaku article carefully.
http://kotaku.com/5961600/if-your-wii-u-wont-connect-to-your-...

If it doesn't work out and you are certain you followed specifically what the instructions have said, contact Nintendo.