If you can access the internet in safe mode, but not on the normal desktop, it would be a driver issue. Try downloading the latest driver for your network adapter, then remove the device from control panel's device manager and reinstall the driver from the file you downloaded. Safe mode uses minimal, and in most cases, native drivers, while windows uses whatever was installed for the device.
You can also try going to the device while in safe mode with networking (Again, in device manager) and right click the device and select update drivers/update automatically to see if there is a newer version available through the automatic updating process. Otherwise, finding the model specific drivers and reinstalling is your best bet.