This is the one instance where you have a relatively good chance of resurrecting your drive if you're prepared to put in some time and effort. If the drive does absolutely nothing when you apply power to it (no noises at all), it is often a PCB problem.
With older drives, you could sometimes find a matching PCB from another matching drive and swap it over. Simply swapping the PCB with a matching, working equivalent on a new drive has almost no chance of working and can be outright dangerous to your data.
There are two main causes of failure here, either a TVS diode (fuse to protect your drive in the event of a power spike) has shorted due to overvoltage, or a vital component on the PCB has failed. If you accidentally plugged in the wrong power adapter to your external drive, or you experienced a power surge, a TVS diode might have sacrificed itself. If the shorted TVS diode is the only casualty and the rest of the PCB components are OK, then simply removing the shorted diode is enough to bring the drive back to life.If the TVS diodes don't smell burnt, then the problem is the PCB itself.
A replacement PCB is required, but not just a straight swap. There is an eight pin ROM chip on most PCBs that contains unique firmware info that is required to start up the drive. This needs to be moved from the old PCB to the new in order for the replacement to work. Some hard drives, especially Western Digitals, do not have this eight pin chip;the firmware is stored in the main controller which is virtually impossible to move.
If you want to replace the PCB then you'll need to find a matching replacement and have the ROM chip moved. There are many online providers that will sell you a matching PCB. Some of them even offer to move the ROM chip for you, saving you the hassle of soldering and possibly damaging the chip. If the PCB was the only damaged component and the drive's internals are OK, then after the replacement and ROM swap, your drive should be up and running again.
Another PCB-related item to check are the head contacts. Sometimes they corrode with time, but are easily cleaned with a rubber eraser.