The “ROM” in a phone which holds the OS is actually PROM. Which is to say a device whose bits can’t be re-written to individually, as can be done on a disk or RAM. Instead the entire thing has to be erased all at once and then completely re-written from end to end.
And the engineering term for that *clear all & rewrite *process is “flash”. And has been since PROM was invented in the 1970s. Some early PROMS were erased by physically exposing the silicon to a bright flash of ultraviolet light which fried all the trapped electrons out of the cells holding "1"s, so they all became "0"s. That erased the whole thing in a flash, then the new data could be written into the newly-blank device.
The ROM in a PC that stores the BIOS is also really a PROM. Which is why you “flash” the BIOS of a PC to update it.
Now in a PC the OS is a big bunch of software. With lots of configurable options chosen during set-up. And which parts of the overall OS code actually ends up on the hard drive depends on those options and on your particular hardware configuration.
The term “install” was chosen for the process of configuring and transferring software, OS or otherwise, from the distribution media to the PC’s permanent storage, typically a disk drive or SSD equivalent. And this term came in turn from the same practice used on minicomputers before PCs, and mainframe computers before minis.
So at the end of the day, we have the phone and tablet terminology bubbling up from the underlying hardware, through the hardware-abstraction-level software, and the OS-level software, and the installation control software all the way to the UI & marketing level.
Whereas in the PC (or Mac) case the terminology comes from higher in the stack. It starts at the OS-level software, bubbling up through the same higher levels: installation control software, UI, & marketing level.
And that’s what’s really going on. *Why *is it that way? Essentially arbitrary accidents of history.
When phones started out, they were pretty dumb devices not intended to be updated during their life. As they slowly morphed from single-purpose devices implemented in fixed-software-on-hardware to full up PCs in your hand, some terminology has remained rooted in their origins and other terminology has kept up with the changes.
We really ought to stop calling these devices smart-phones, because voice telephony is one of their least important and least complicated functions. They’re handheld computers. My vote for the new term is “tracker”, as in the device everyone and his Big Brother use to track me. And you.