SUDDENLY Mexico and Southern Italy were blasted into an arena, where only the organised crime groups in each region can decide who is victorious. 

Mexican Drug Cartels vs Italian Mafia

In this situation, who do you think would win? Of course, these groups operate in different ways, but I think they are comparable enough to at least attempt to make a debate.

So... it's just the Mexican drug cartels and the Italian Mafia in an arena type setting?

I would guess the Mexican drug cartels are far better armed.

Now, put them in the same city, or try and invade each others territory and things open up more for the Italian Mafia since they seem to work a lot better with the authorities and are better at logistics and organization.

Well... assuming by Mafia you mean ALL of the sothern Italian crime families, and not just the actual mafia which only has to do with Sicily.

Straight firepower and Brutality the mafia doesn't stand a chance, give them the ability to bribe officials, get law enforcement to crack down on certain areas, portect and blackmail people, get extra money from running rackets better and more efficently... the Mafia have a  good chance.

Coincidentally, the Mexican drug cartels and Mafia apparently work together to move drugs like Cocaine into Europe.

The cartels would have the mafia outgunned. The Obama administration would see to that.

Wasn't this on Discovery Channel? Deadliest Warrior.. I know there was an episode of Yakuza vs Mafia..

Oh and the Russia Mob would beat both in both kinds of scenarios.

Russian mob both has access to all kinds of crazy weaponry and are great at bribing and working with officials in a constructive manner.

It was all Southern Italian crime families (to make it fairer), and it's not a straight-up fight, the civilian population of these countries still exist to be manipulated in whatever way they choose