|As per my #2, I think you ought to take a peek into Device Manager to see whether there are any errors showing. In Safe Mode there are only a few drivers, which could be the reason Internet works from there, ie. there might be a driver issue which shows up in Device Manager.|
You can get to Device Manger various ways but for speed just put devmgmt.msc (note the dot) in the Run box and hit Enter. Errors show up with red or yellow symbols. If one is showing let us know where it is. If everything looks OK we can persue the suggestion given in #1.
Ms Hulot's boat