One thing you tend to come across quite often when you start obsessively searching for anybody's opinions and/or comments about Blake's 7, is the question of whether the Federation in the B7 universe (The Terran Federation) is any relation to that of the Trek universe (United Federation of Planets).
The immediate similarities are quite obvious - they're both referred to as 'the Federation' for a start. And then there's the symbol, which is pretty much the same, except that it's been rotated through ninety degrees (in case anybody hadn't noticed). This is enough to start an interest, and from then the general feeling of 'wouldn't it be cool if the Federation did go all evil and started doing mind control experiments and stuff' takes over. It's this kind of thing that causes crossovers to happen. And those episodes of DS9 with Section 34.
Not that there's anything wrong with crossovers. They're fun. In fact, the main problem I have with them is not knowing anything about the other fandom, which rarely stops me reading them.
But I digress. Firstly, let's deal with timescales. From the information given in the series' we have to assume that the Trek Federation (UFP. I'm getting sick of typing federation) came first. We have a concrete date for that. TOS was set in the twenty-third century, TNG, DS9 and Voy in the twenty-fourth. We know a significant amount about the events which take place between their time and ours, and there was no Terran Federation, nor was there time for one to develop.
There was never any specific date given for when B7 was set. I always got the impression that it was in the very distant future, which coincidentally ties in with the above conclusion. In the B7 universe it is a myth that all life originated on earth, indicating that either the human species has existed for long enough that concrete evidence about their origins has been forgotten, or there's been one mother of a cover-up. Compare this to the Trek universe, where all the humans are quite knowledgable about their origins.
Also in the B7 universe, there are colonies of humans spread all over the galaxy, it seems. Some have reverted to barbarism, some have highly advanced technologies, but they are all recognisably human. Even those which are described as alien (the Auronar, the System), are more probably offshoots of Earth which have simply experienced cultural drift. Compare this to the Trek universe, where the galaxy (what little of it has been explored, at any rate) is populated by a host of different species. It is doubtful that, no matter how evil the TF is, it has managed to successfully wipe out all traces of these cultures. Especially since it's having a bit of a hard job keeping humanity alone in check.
Add to that the fundamental differences in the humans portrayed in each series. In Trek, Earth has become a paradise, humanity lives (mostly) in peace and harmony, and the species as a whole is dedicated to bettering itself intellectually. In B7, humans are paranoid, untrusting (or worthy), manipulative and usually only concerned with their own immediate wellbeing. This could simply be the results of the same culture separated by a long period of time, but I'm going to assume it isn't, because that way I have a point.
Although it undoubtedly would be very cool if the UFP mutated into the TF. Because Star Trek, much as I love it (well, like it. Most of it. Some of it. Sometimes), is just far too nice. And everything always turns out okay in the end. I would love to think that humanity's attempt to spread love, justice and the American (did I say American? I meant Federation) way throughout the universe ended so pitifully.
Anyway, in conclusion. In my humble little meaningless opinion, the only link between the UFP and the TF is the B7 production team (or whoever) pinching a logo, and hoping nobody would notice. And B7 is preTNG (I think. I could be hideously and embarrassingly wrong, but that's what happens when you can't be arsed to check dates) so they might have had reason for assuming it wouldn't be as painfully obvious as it turned out to be.
Oh, and another difference. In the B7 universe, everyone speaks English or French. In the Trek universe, everyone speaks English or Klingon.
or...