There is one notable similarity between the two settings. In both, Britain has lost its monarchy. In the Brigade Leader's world of 1970, we are told that the royal family were all shot after the second world war, and the dictator of the Republic appears in photographs on every wall (in fact depicting the BBC's chief visual effects designer, Jack Kine). In Pete Tyler's world of 2006, the "President of Great Britain" is played on screen by Don Warrington. We are not told what happened to the royals in this case (or to Northern Ireland).
The Doctor's reaction in the Brigade Leader's world of 1970 makes it quite clear that the fate of the royals is a Bad Thing about this world, one of numerous early signals that it is a much nastier place; this is backed up by the fact that almost every sympathetic character in our world has a nastier counterpart in the Brigade Leader's world, with the exception perhaps of those who were already nasty on our side. At that time, of course, dictatorship was a lot closer to the audience in both time and space. The second world war had ended twenty-five years earlier, if not in the living memory of the kids in the tea-time audience, certainly in the memory of most of their parents. Portugal, Greece and Spain were still dictatorships, and Communism still had almost two decades to go in eastern Europe. The royals were seen as a contributing factor to Britain's political security. (Gosh, it's difficult to remember those days; indeed, the fall from grace of the royals and the fall of Communism happened about the same time.)
In 2006, Pete Tyler's world at first looks rather more attractive than ours. The very first thing we see are Zeppelins, which, let's face it, are just cool (it's a safe bet that very few in the audience could remember their deadly use in bombing raids nine decades earlier). There are mildly interactive billboard advertisements; the president is Afro-Caribbean. Nice people who are dead in our world (Pete Tyler himself; Micky/Ricky's grandmother) are alive here. It seems like the Doctor and friends have come to a better place - but it rapidly becomes clear that we haven't; the President tells Pete that it is "a sick world", his government is in the pocket of an insane plutocrat who also controls the media, the homeless are being rounded up for scientific experiments, working class areas are under a curfew, and even the rich have to get hold of whisky though the black market - "pardon me, Mr President!" (though given my earlier suspicions about Northern Ireland, that may well be whiskey).
In the Brigade Leader's world, the violence of the system is overt - the Doctor's first exposure to it when the Republican Security Forces start shooting at him, he leaves it as Squadron Leader Liz Shaw shoots the Brigade Leader dead, moments before they are all engulfed by magma from the Earth's core. Pete Tyler's world, however, finds redemption rather than holocaust. In "Doomsday" we find out that the "People's Republic" have taken over Torchwood and put him in charge of it, in contrast to our world where it is a law unto itself.
So, what do we learn from this? One thing that I think hasn't changed is the acceptability of parallel-world storylines. The notes on the "Inferno" DVD point out that the 1970 audience could well have been familiar with the 1969 Gerry and Sylvia Anderson film, Dopplegänger aka Journey to the Far Side of the Sun, which featured a similar storyline, and the John Wyndham short story "Random Quest", in which the central character is shunted into a world where the second world war didn't happen, was filmed for "Out of the Unknown" that same year (directed by Christopher Barry, who also directed the first Dalek story for Doctor Who). Indeed, if anything I felt that the explanation of what was going on was a bit more painfully explicit in 2006 than 1970.
The political difference: the threats to Britain's social order are perceived today as much less visible and more insidious than they were in 1970. Then, the enemy was visible and armed, defending scientists who were likely to bring us all to Hell. Now, they are more subtle - corrupting our own government in order to remove our brains.
Scary, isn't it? But that's the point of good drama.