IBM still develop and sell tape drives today. The capacity of them seems to be on a par with today's hard drives, but the search time and transfer rate are both significantly lower than that of hard drives.
So when is tape drives preferable to hard drives (or SSDs) today?
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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.