Although both styles of Tango use the same name, that is where the similarities end. Everything else is different.
The Tango comes from Argentina where it was developed in the slums and brothels of the capital city. In was a dance for the lower classes and not danced in society.
Then Tango was then taken over to Europe where the social classes started to dance the Tango and changes started to occur. It was in Europe that the Tango evolved into what is known today as the Tango that is now part of the Ballroom competitive discipline.
Its style changed far away from its original roots when it was danced in Europe. This includes the steps, posture, poise, position, timing, footwork and even the music started to change.
This is why when you look at the two different tango styles today they look totally different. You also have to consider the fact that dance styles also continuously evolve so even today both styles of tango are evolving and moving into different versions of the original form.
WHICH SHOULD I LEARN TO DANCE?
Both styles are great in their own right but I think the real question is what your intention is. If you want to get involved in the competitive world of Ballroom dancing then you'll have to learn the Ballroom version of the Tango.
If you're interested in taking grades and medals with dance societies such as the IDTA or ISTD, although you can do this in both types of Tango it is the Ballroom Tango that is more suited to the grading system.
If you would like to dance socially at dance events then there are usually more Argentine Tango events for dancers than there are Ballroom or Latin events. Both styles of Tango are great to dance and offer opportunities for the dancer.