How is an overlayfs different from just mounting another disk/partition over a directory?

I have OpenWRT installed on some of my routers and to add additional storage for settings as well as programs that might be installed on the router and maybe logs, OpenWRT recommends you plug storage into it and use an overlayfs. I also have a SBC where I just mount an external drive overtop of my home directory on boot to store the home directory externally off of the SD Card that the bootloader and OS are installed on; since the storage on the external drive is more reliable than the SD Card, despite running slower. What is the difference between these two strategies? They are both basically Single Board computers with Linux, and when the external drive fails to mount, in both cases we're left with a directory full of the content of the original directory, where the drive would have been mounted before. The only think I can think of that is different, is that the settings directory for OpenWRT (/etc) is being mounted on the external drive, where this is not the case on the SBC.