Just because tapes seem outdated to the average guy who has no clue on how IT really works doesn't mean they actually are. Some companies have to work with or produce such massive amounts of data on a daily basis that cloud backups or stuff like that is not an option. Clever (read, distributed) server farm design can minimize the problems of outages and data loss to some extend but nothing beats having a complete copy of your stuff on tape. Of course the key here is to choose quality tapes and decent drives, set up a proper backup (and most importantly VERIFY) system and store the tapes in air-conditioned safes on different locations, a bunch of rotten old DAT tapes scattered across the administrator's desk is not the way to go.
This is where most of the problems with tapes and data backup as a whole come from, there still are too many admins out there who play Russian Roulette with their system and give a rat's ass on proper backup strategies... ...until the day their ever-so-important logon server crashes. I've seen companies go bankrupt because of a massive data loss and zero useable backups. Sure, a good backup strategy costs a lot of money but when something really bad happens, it pays off in the end.
Oh, and about that 80s hipster bullshit, a lot of music studios still use magnetic reel-to-reel tape machines for recording purposes. Of course they're not comparable to the good ol' cassette tapes we all used back in the day (well, most of us old geezers anyway) but the basics are the same. The answer lies in this thing called tape saturation which will give many recordings that extra punch that is almost impossible to obtain by digital means. Tape saturation is an effect of good-quality magnetic tapes, if you record something on a large-format tape and crank the gain up, the result is not clipping as in most digital systems but a warm overdrive, a sort of natural compression that leads to a fat and in-your-face sound.
