In the 2008 version, humanity is on the brink of annihilation. Impeded only by a handful of brave Viper pilots, swarms of Cylon Raiders continually threaten to wipe out what remains of the fleet. Heavy Raiders bring Centurions onto Galactica that march inexorably towards the ship's vital systems, and the fearsome Basestars ceaselessly pump out reinforcements as their guns pummel Galactica unless silenced with a nuclear explosion. All the while, internal politics exacerbated by hidden Cylon agents threatens to tear the fleet apart from within as the humans struggle desperately to hold it together long enough to reach Kobol, the signpost to Earth.
In the 1978 version, there's a single defenseless and inanimate Cylon Raider that gets dragged around a lot. If you drag it into your corner, you win! All the other Viper pilots are trying to stop you for some reason by blasting you to pieces. Perhaps all the pilots are very drunk.
In the 2008 version, you can play as William Adama, Starbuck, or perhaps the enigmatic Gaius Baltar, any one of whom might turn out to be a Cylon agent secretly working to destroy the fleet.
In the 1978 version, you can play as blue.
The 2008 version uses a simple d8 to resolve ship combat (among other things), but the heart of the game's conflict is in the skill check system where players contribute cards to a face-down pile to deal with various crises. In each check some colors of cards will be helpful while others are harmful, and the cards are shuffled before they are revealed so it is unclear who has aided the humans and who has sabotaged them. The Cylon players will need to be subtle and choose their moments if they are to successfully destroy the humans' chances of survival without arousing too much suspicion.
The 1978 version has arguably one of the most ridiculous combat mechanic ever conceived. By playing a "laser torpedoes" card, you get to spin the spinner. The number result (from 1 to 6) tells you how many further times you get to spin the spinner. Each of these times, you check which of four colors the spinner is pointing to, and all the enemy Vipers in that quarter of the board get shot. By playing one laser torpedoes card, you could realistically shoot all, none or some of your opponents. Who knows? At least the mechanic is entertaining, in a putting stuff up your nose kind of way.
I would recommend the 2008 version to anyone who doesn't mind a somewhat longer and more complex game. The only real gripe I have with the game is that it can go upwards of two hours which keeps it from hitting the table as often as it would otherwise, but I still have a great time every time it comes out. Extra recommendation if you're a fan of the show, or enjoy an excuse to lie to peoples' faces.
I would recommend the 1978 if you have a sense of humor and would like to compare just how far boardgames have come in 30 years. Extra recommendation if you're a fan of Plan 9 From Outer Space, or anything else that sucks in a brilliant way.
In conclusion, the series finale was somewhat disappointing.