Actually, there isn’t any difference in meaning. Both words apply to paralysis of all four limbs. And both terms are used interchangeably these days.
The difference is in the derivation of the words. The word “Quadri” means four in Latin; the word “Plegia” means paralysis in Greek. So the roots of the word “quadriplegia” which means paralysis in all four limbs, come from both Latin and Greek. It combines two different languages.
The Greek word for four is “Tetra.” Combine that with “plegia” and you have a word with Greek roots for both halves. The British have always used the term “Tetraplegia” for four-limb paralysis, so they are not combining Latin and Greek words.
Such distinctions are important to the English, but Americans don’t seem to mind. Although there was a movement in the 1990’s to try to adopt “tetraplegia” in America, it never really caught on outside of the medical literature.
That’s why most Americans still continue to refer to “quad rugby,” for example, and why the word “quadriplegia” remains in common use.
Incidentally, since “para” is the Greek word for two, and “plegia” is Greek for paralysis the word “paraplegia” all comes from the same language of origin—Greek.