Not a huge amount of difference in the end product to be honest.
As RaD posted, the main "functional" difference between the two materials is how they are heat treated after welding. While I'm not a metalurgical engineer, my understanding of "why" the materials are heat treated after welding differs for both 6000 and 7000 materials.
The main advantage of 7000 material is that is requires only an artificial age after welding. The material loses only minimal strength after welding. However, it is very subseptable to stress corrosion cracking. If it is not properly aged after welding, the frame's long term durability will be significantly reduced. Almost all aluminum alloys age somewhat at room temperature. However, I'm pretty sure that you MUST artificially age 7005 aluminim post weld to maximize its resistance to stress corrosion cracking.
Because the frame only needs to be "baked" after welding, it makes for a much easier manufacturing process. Any builder can invest a small (relatively) amount of money and build a heat treat oven. Typically, only two different temperatures are required, and they are fairly low (the high 300's and mid-high 200's F).
So, a welder can weld the frame, straighten it, then stick it in an oven and take it out 11 hours later (or what ever the time is).
7005 is one of a few 7xxx alloys that can be easily tig welded. My understanding is it was invented precisely for this purpose (i.e. easy welding and lack of a solution / quench process - explained below). I'm not 100% sure about this, but otherwise, there's no real advantage of 7005 as it is typically much weaker than other 7xxx high strength alloys such as 7050 and 7075.
One last problem with 7005 is its severe lack of availability in North America, especially as of earlier this year as Easton stopped tube production in the USA and moved it overseas. So, if you're just buying the basic pre-made pieces for a road bike, then you're going to be ok. However, getting material for for CNC'd parts and such is now very difficult (and the material is very expensive).
6000 series alloys (6061 is the most poppular) is a very common alloy and typically called "air craft" grade aluminum (there there are many different types of aluminum used in aircraft).
It welds easily and material is very very common and generally easy to find, which makes it easy to design and manufacture complex CNC'd parts. It is very soft when annealed which makes it easy to form into complex structures like monocoque shells.
It's slightly less strong than 7005, but has a much higher durability because of greater elasticity. It also has better corrosion resistance. However, it requires a two stage heat treating process that is much more complicated than what 7005 requires.
When welded, 6000 alloys lose significant strength (i.e. annealed 6000 is about 10-20% of the strenght of a T6 heat treat grade of the same material). So, the 6000 frame must be heat treated, otherwise, you'd bend it just by pedalling it.
The first stage involves a solution bath typically around 800* F for typically 30-60 minutes. Then, the frame must be rapidily quenced: almost down to room temperature within less than a minute I think. This locks the alloy's molecular structure in place. Afterwards, the frame is re-aligned, then
it goes for an artificial age.
The solution process redistributs the alloying elements in the frame, the quench looks them in place. Afterwards, the aging gives the material it's high strength.
Functionally, very good bikes can be made from both materials - the 6000 or 7000 decision is typically a manufacturing decision and has very little to do with performance characteristics. The ONE exception is Scandium material for XC and Road bikes. This material achieves the same stiffness as 7005 aluminum, but allows for thinner wall sections (I think) resulting in a weight reduction of the frame. However, it's not suitable (or practical) for heavier duty frames.
Hope that helps and I hope that I have everything more or less correct.
Edit: RaD - that link is cool - thanks for posting it! Another cool link is www.matweb.com
if you're interested.