Connected to wifi successfully, but no internet connection
You've provided great troubleshooting info, we can isolate the problem to being your Mac and not your router, we can isolate the problem to OSX and not your WiFi card, etc. 

I want to say it's an IP address problem. Maybe Windows has taken the IP address that your router gave your Mac's MAC address. I'd recommend going to system preferences --> Network. On the left column does WiFi say "connected" or "self-assigned IP" or something else? In the mean time, I'd recommend going to system preferences --> Network--> select WiFi on the left column --> click advanced on the bottom right --> TCP/IP tab --> Renew DHCP lease. 

If this doesn't work, you can try on the WiFi tab (instead of TCP/IP), select your WiFi network and then click the minus (-) button to remove it, click OK, and try to reconnect to the WiFi network again. 

If that doesn't work, you can hold option and click on the WiFi symbol on your menu bar (top right). Does your IP address look right? If you're not familiar with networking let me know what your IP address and router address are and we can go from there.