Any personal computer built in the last five years can do anything a smartphone or tablet can in terms of processing power and connectivity. A smartphone is a computer with hardware that enables it to use cell phone networks and make calls.
My inner cynic strongly suspects that Facebook and other similar corporations really like the control they have on the overall user experience on the two major mobile operating systems; i.e., eyeballs on a smartphone or tablet are worth more than those on a general purpose computing device.
Too much freedom on a personal computer; with browsers that feature all kind of privacy enhancing add-ons such as ad-blockers and tracker-blockers. Much harder to monetize.
So are you saying that a desktop PC's hard drive can't handle storage of some text messages but a phone can?
For consumers (and facebook is a consumer company) it's all about who owns mobile (and can also compete with facebook). They want desktop to just be enough of a feature to be more appealing than other platforms, but not enough that it detracts from mobile.
Whatsapp don't retain messages/media after they've been delivered to your phone, which is a compelling privacy feature for many.
It's also what allows them to serve such an enormous user base with limited hardware. Their technology stack (FreeBSD/Erlang) is pretty interesting, more info here:
2014 talks by Rick Reed:
https://www.youtube.com/watch?v=c12cYAUTXXs
https://www.youtube.com/watch?v=TneLO5TdW_M
Slides:
http://www.erlang-factory.com/static/upload/media/1394350183...
There's likely no technical reason why you couldn't use a pc instead of a phone for users that want to use the pc as the primary client (with the phone optionally accessing the DB on the pc in the same way that the desktop client does for the phone). Perhaps they've decided that this is a small and declining market.
Edit:
Slides for second talk
http://www.slideshare.net/iXsystems/rick-reed-600-m-unsuspec...
440k connections/sec, 1.1 million msgs/sec, 1 billion images/day, and that was in 2014...