Drivers work on OS level. For example if you had machine that runs both Windows and Linux you need to install drivers for each of those OS separately. BIOS update works on hardware level, once it is done it works for all systems you run on given PC. Difference here is crucial: you should keep DRIVERS up to date (meaning you should install new versions when they show up); but you should ONLY update BIOS when you have a problem that such update is supposed to resolve. Why? Because updating BIOS is risky - failure to do so properly or sometimes just bad luck can cause update to fail, and this usually ends in bricked board (meaning you have to buy new one).
Now, back to your problem. If it was working and now is not, it still could be caused by drivers, but only if Windows has changed them in meantime (Win 10 can do that). But anyway in such situation your first step is to reinstall newest LAN drivers just to make sure it's not them. If it won't help, you have to check if cable/router if they still work fine. If still nothing, you are facing probable motherboard hardware failure.