Idiot's guide
To get a simpler view on this tyre (tire, US) explosion business, I find its much easier to work with numbers as well, rather than word equations alone.
Example:
Tyre pressure gauge (gage, US) indicates tyre is inflated to 600 lbs/square inch.
Outside (or ambient) air pressure gauge indicates 15 lbs/square inch on the ground.
Outside air pressure gauge at a typical airliner cruising altitude indicates 2 lbs/square inch.
Actual tyre pressure=Indicated tyre pressure - outside air pressure
Tyre pressure on ground=600 - 15
which equals 585 lbs/square inch.
Tyre pressure at altitiude=600 - 2
which equals 598 lbs/square inch.
So the total difference between the pressure on the ground and at altitude equals 13 lbs/square inch (15 - 2).
The tyre is only therefore increased in pressure by 13 lbs/square inch, which is not a lot, so the tyre is not under enough strain to burst or explode.
I hope that has helped!
Cheers, Tri