As far as I know, the obvious difference comes down to depravation.
If you have a computer that does all sorts of important things, like work, or even playing games, then exploring how it works isn't the obvious first thing that comes to mind for interacting with it.
But if you have a computer that doesn't really have any use-cases that you personally care about, exploring how it works is pretty much the only thing you there is to do, when playing around with it.
Similarly, if you break your computer while exploring it, and you have plenty of other things to do, then you'll maybe give up on your explorations.
But if you break your computer and you have nothing else to play with, you'll see the breakage as an obstacle in the way of your continued entertainment—sort of like you might see the Internet being down—and try to figure out how to get it back up and working again. You might not succeed, but it's the attempt to get it working that builds the "engineering mindset", not the success per se.
I'm a programmer today mostly because: my parents weren't always around to bug; I didn't live near-enough to friends to always have them over or visit them; and I didn't own any video games, but merely rented them (mostly timed to when said friends came over.) Then, when I was 9-or-so, my family got a computer that ran Windows 98, had no Internet connection, and had no games installed. I "played" with it by poking around in MS Paint and discovering the hidden sets of system MIDI files; but also by doing weird things like making "Briefcases" just to see what they did; or creating desktop shortcuts to nothing-in-particular just so I could put some of the "cool" icons from pifmgr.dll on my desktop, as if I was collecting game power-ups.