What are the advantages of tape drives?

For me, the single biggest argument in favour of tape is that doubling your storage capacity is cheap. That is, to go from 1TB of HDD storage to 2TB is the same as going from nothing to that first TB. With tape, you pay a large premium for the drive, but storage after that is comparitively cheap. You don't have to have lengthy budget meetings about increasing the size of the storage NAS by 15TB, you just order another box of LTO5s.

(Chopper makes a valid point about compulsory labels, but tape labels are in a standard format, and there are free software solutions to printing your own onto label stock.)

Tapes are much easier to ship, and easier to store, than HDD and HDD-like media. They're more resistant to shocks, and their temperature tolerances are higher.

They also benefit from the existence of autoloaders. This allows you to spread a large dump over multiple storage containers, which means you don't have to worry about how to break up your backups. While it's perfectly possible to make an autoloader for HDD-type media, I've never seen one, and I suspect the lack of standardisation in physical package size will make it difficult to bring one to market at a reasonable price.

Your point about transfer rates is valid, but in the context of backups it's of minimal import. The time required to back up a 1TB file system to anything is large enough that you shouldn't be doing it on a live file system; and if you're dumping a snapshot to tape, who cares if it takes an extra hour or two? Search times are an equally minor concern, because all decent backup software maintains indices, so one can generally go straight to the relevant portion of the tape to restore a file.

Edit: after an incident earlier this week, one more advantage of tape has struck me most forcefully. A client got infected with ransomware, which promptly encrypted several hundred gigabytes of their main corporate file server. Online backups are all very well, but any system that can write to those backups can rewrite or erase them as well - even if you would rather it hadn't. That certainly doesn't argue against all HDD-based backup systems, but it is a weakness in the simple "let's just have a big NAS and do all our backups there via cron" approach.

My client has tape storage, by the way - so apart from a couple of lost days, no harm was done.

Data retention - if you put a disk and a tape with the same data on it in the same physical location it should last a lot longer on tape than on disk - potentially by an order of magnitude.

Also tapes are generally better at dealing with the rigors of being shipped from primary to secondary/off-site data centers multiple times.

Tapes don't compare with disk quite like that. Tapes are for backing up, and they compare more with deduplicated and replicated disk like Data Domain, or optical media.

The main reason for tape backups is that it's cheap. You can afford to store 10 full copies, even though you don't really need them, because the media is so affordable. The next main reason is that it's easy to offsite the data. Hard drives don't travel (or jostle) well. Tapes do, for the most part, and can be brought offsite. Offsite full backups can also be accomplished with things like replication, but that's a lot of money to your ISP.

Really, boiling it all down, it comes down to cost. Tapes are cheaper than their competitors or it wouldn't exist.

In addition to all the above, there is generally "less stuff to fail" with a tape. It's a very simple device, and you don't have to worry about the onboard electronics (the drive controller with a disk, even SSD). Additionally, their storage and transport requirements are generally a bit looser. They can be subjected to heat and humidity swings as well as acel/decel forces that normal hard drives just cannot.

Lastly, there are sites that have enormous infrastructures built around tape storage, with many tens of thousands of dollars sunk into tape silos. We can assume these devices will continue to operate for a while, but drive interfaces change yearly. A DDS3 or DLT tape I wrote in 2001 I can still find a drive for; this is not quite the same story for old U320 SCSI.