Tango is an idea, specifically an Argentine idea, and so anybody who mentions tango in the same breath as
ballroom dancing is not thinking about tango in the Argentine sense. There are three dances considered tango:
milonga,
tango waltz, and tango proper. Well, five: there were
candombe and
canyengue, too, in the old days.
Milonga is sort of a "one-step" dance, set to peppy music with a steady beat. The general rule is to step on every beat of the music.
Tango waltz (or "vals" as some people spell it) is tango steps danced to waltz music (that good old "oom pah pah" rhythm), so it sort of looks the same as tango, but it has a more fluid feeling. Tango orchestras often play a few waltzes, and sometimes they're sentimental, but the ones I like best are by the D'Arienzo orchestra, and they're pretty snappy and even tough.
Tango proper is more or less derived from milonga (and through that from candombe), and canyengue. Tango music is often complex and lends itself to varied interpretations in the dance. You generally step on every other beat (by the way, the slow-slow-quick-quick-quick business is an invention by outsiders), but half the fun is changing the rhythm of your steps. The other half of the fun is stepping into your partner's space (sacada) which looks like you're kicking their foot out of the way. The third half of the fun is the close embrace; it looks like you're leaning on your partner, but actually each person is balanced and standing up separately. It's a lot of fun to do fancy turns or make your partner kick their foot up (voleo), but the main thing is intimacy: it's really good to be close to someone.
I'm not generally opposed to people borrowing stuff from other countries and changing it. Hey, it happens all the time, with fantastic results. My favorite example is Costa Rican swing, which (as far as I can tell) is sort of a cross between lindy hop and salsa, danced to cumbia music. But the sad fact is that Arthur Murray, et al, have totally mangled tango, by obliterating everything that's beautiful about it, and introducing ridiculous dross like head snaps, syllabification, and an emphasis on competition. I guess they're free to do as they want, but I wish they'd call that dance by a different name.