1. more address space...
the first notable and obvious benefit, is for the entire netizen community, more ip addresses to accomodate more devices now and into the future  thats one problem we can forget for now.
2. faster latency
You can run a test here
http://ipv6-test.com/pingtest/

well latency is most important for gamers. I remember a few years back when i was playing world of warcraft with Tmnut Adsl to the US server. I was averaging 600-800 MS latency. Barely playable as my reaction time was slower than other players, which is most noticeable in the battlegrounds and pvp. Basically i was playing with a nerf around my shoulders in terms of high latency 

Then later i got Unifi 10mbps. My latency to US server these days around 200-300ms. This is drastically better 
I suspect they achieve this by getting around the whole NAT deal in the router 
VOIP also benefits from lower latency. If you ever use teamspeak, Ventrilo, Mumble, skype..... you may have at some point come across latency issue, like delayed time before the message gets through via audio So lower latency here means, less lag time between input/output of audio through the internet 
3. improved security
IPv6 has several advantages over IPv4. IPv6 has several privacy enhancements: it has more encryption and authentication options, which ensures that networks are less susceptible to being compromised overall significantly more secure.


4. p2p benefits
In addition IPv6 supports multicast, which saves network bandwidth by allowing bandwidth-intensive packet flows—such as multimedia streams—to simultaneously be sent to multiple destinations.

If that sounds confusing, well a very wise person (Confucious) once said, a picture is worth a thousand words 


But because now is still in the transistional period, there are some slight hickups for the meanwhile
5. Mobile devices retain IP addresses when moving from network to another on the go

Another advantage of IPv6 addressing is that when you're moving from place to place with your mobile device, you'll no longer need to worry with getting a new Internet address at every stop. With Mobile IPv6, whether your smartphone or tablet is connected to the Internet with Wi-Fi or WiMAX, your device should retain the same address.

If the wireless infrastructure around you is up to snuff, mobile IPv6 will let you seamlessly move from one form of wireless connectivity to another without losing your connection or needing to pick up a new IP address.
6. faster internet speed