What happens when a flash drive wears out?

Yes, they do very similar operations when sectors fail: both silently move sectors to unallocated area after failed write verification. Yes, their number is limited on both the memory types.

There is just one, but important, difference: number of writes on hard drives isn't limited, so failure of sectors in hard drive is unexpected and is probably sign of failure in some other it's part: a crashed head or scratched surface or overheated controller etc. Most time you will want to replace the whole driver when bad sectors begin to appear even if controller has remapped them successfully because soon the whole drive will fail.

Failure of sectors on flash drive is expected and (quite) often happens in normal use: a sector has a limited and determined maximal number of writes, avg. 100000 on a recent flash chip (I'm referring to datasheets of common I2C flashes, but the technology is very similar in USB flash drives). After this number for any sector is exceeded it will fail, sooner or later, so it does not refer to overall state of the drive in any way.

Because hard drives weren't supposed to have any bad sectors during normal work, there were not any technology that was able to detect them (and also remap and notify user). Some years ago SMART was created to do this job.

As remapping for flashes is required by the technology, SMART-like technology was used in flash drives initially. No standardized methods of checking it's status exist, and AFAIK there are no vendor-specific utilities too. That's all about cost reducing.

I've had quite a few failed flash drives, and there was no warning - the drive simply stopped being readable. They were large drives, and I didn't put much on them, so I assume things were being written to good sectors automatically, if silently.

Moral of the story? Back up everything on your flash drives.