The number one advantage of IPv6 over IPv4 is the much larger address space. To quote
Vint Cerf, one of the "fathers of the internet", a few weeks ago:
The other thing that is very important for the last decade or so, is the introduction of mobile technology. There are 5.5 billion or so mobiles in use today; not all of them are internet-capable, but probably 20% to 25% are, and over time that percentage will go up. The reason that's important, is that for many people their first introduction to the internet comes through a mobile as opposed to a desktop or a laptop or an iPad. For many it will stay that way, and for others: they will begin to accumulate other devices that they will use in addition to their mobiles. But that adds a huge demand for address space on the network, because if every mobile that is internet-enabled has to have an IP address assigned, eventually you start to run out of IP addresses. And in fact, that's what we're faced with today.
We chose a 32-bit address space in 1973, in order to carry out an experiment. And I want to emphasize that this was an experiment. What I though was that if the experiment succeeded, then we would design a production version of the system. Well the problem is that the experiment never ended, and so we're still using the experimental internet which only has 4.3 billion terminations built into its design. So in 1996 in a great panic, we developed, and here "we" in this case is not Bob (Kahn) and me but rather the IETF, developed an alternative packet format called IPv6. It has 128 bits of address space and if you do the math that's 3.4*10^38 addresses. This is a number only the (?) can appreciate. We are now in the process of introducing in parallel with IPv4 the IPv6 formats. So on june 8th last year (2011), many of us in the internet community turned on the IPv6 capabilities that we had in addition to IPv4. On june 6th this year, all of us are going to turn on IPv6 and leave it on. So this is a very important year for the internet because IPv6 will be launched on june 6th. Google and many other have been preparing for this for several years and we're all looking forward to seeing what happens when we turn it on and leave it on permanently.
On a more practical note: bcause of IPv6 much larger address space, addresses are much cheaper, and typical home internet connections are intended to provide many - enough to give each household device its own public address (though firewalled) without needing atrocities like NAT (creating a separate independent local network, and translate addresses between them) and UPnP to overcome it.
