All "generic" hard drives that meet the specifications should be fine. I recently bought a Western Digital and had no problems using it on my PS4. The difference isn't in the hardware, it's in the data. A PS4 hard drive will be "formatted" in a special way that cannot normally be read by PCs or Macs, similar to how some Mac-formatted drives can't be read in Windows, either.
Once you format it as use for storage on a PS4, you won't be able to use it on your computer without formatting it again and losing all your data. You can also use it as normal media drive if you just want to have a place to store all your movies and audio files. In that case, you can leave it formatted as a PC-compatible format, such as exFAT.
However, do be aware that some models may include extra features, like full disk encryption, a boot partition, or some other unusual configuration. Those drives may or may not work as expected on a PS4. I remember having one flash drive that had two "drives" within it, one for portable software and another for storage. I'm pretty sure that wouldn't have worked on a PS4. However, the typical baseline models without any extra bells or whistles work just fine.
Usually something like a "game drive" will have been specifically pre-formatted for use on a PS4, so as to save you the time of doing a fresh format. However, that's mostly just a gimmick, since it only takes a moment to format a drive anyways. If you find a really inexpensive PC-compatible drive, go ahead and use it on your PS4.