Wow... where to begin. There are advantages to each, people swear by each, and so it's largely going to depend on the driving and creature comforts you're cool with. If you go the 80 route, check out slee offroad's website to see the variations over the years (you'll probably want a mid-90s 80, I think, if you chose that model).
I have a 62 and I love it. The style, the slightly rough ride (esp. when compared to the 80), the ruggedness, etc etc. I smile everytime I see and drive it, which I can't say would happen with too many other cars. I have learned to love the quirks and even the upkeep that it needs. I love the fact that even I, a newbie wrencher, can figure out where most parts go and how to replace them just using the FSM.
That said, the 80 seems to have cheaper upgrades, easier to find parts, and just as capable offroading abilities with an uber-comfortable ride. Depending on what you want, you might actually spend the same amount of money buying a pre- locked 80 with suspension upgrades than if you were to do those upgrades to a 62 and the ongoing maintenance that a typical 62 requires (everything adds up)... but it's up to you.
I don't want to simplify it by saying it's a style issue, but they're both great vehicles. So it kinda is. And as 80s come down in price, the $$$ issue seems to lessen more and more. Putting an OME lift on a 80 is a lot cheaper than it is for a 62.
Try to test drive both... you can't lose either way but they are quite different rides. You'll know which one appeals to you..