On a practical level, IPv6 is easier to administer since in a lot of ways it's more straightforward. For home use it's great since every device gets its own globally unique address (I can `ssh home.example.com` from any IPv6 enabled host).
Firewall rules become simpler if you have a common prefix vs lots of individual IPv4 addresses.
There are already lots of users who have IPv6 enabled: http://www.google.com/ipv6/statistics.html: TL;DR: over 2% of Google's traffic is over IPv6. Running dual stack servers lets you reach more customers.
I guess to an extent asking what people use IPv6 for is similar to asking what people use IPv4 for. You don't normally use IP. You use applications that run on top of TCP/IP or UDP/IP. Having a public IP address makes writing and managing those applications easier.
Edit: Also IPv6 addresses are cheap. An additional IPv4 address will run you about $1/month with a lot of providers. A billion IPv6 addresses is one billionth of the minimum allocation you will get for free. So in the long run, the cheapest DO droplet might cost as little as $4 instead of $5 :).
The idea is that you want a dual stack environment where IPv4 and IPv6 coexist. Mac, Linux, and Windows all support this and do the right thing: if IPv6 is available on the client and server they try that first, if not they use IPv4.
Since you are asking I am assuming your ISP does not provide native IPv6. If you have a public IPv4 address that does not change more frequently than every several hours, the best solution is to set up a 6in4 tunnel (ignore any docs that refer to 6to4, different beast). Sign up for an account at TunnelBroker.net and create a regular /64 tunnel. What this will do is that your router will encapsulate your IPv6 packets inside IPv4 packets and send them to a Tunnel Broker router. That router will then strip the IPv4 header and send the packet off as a pure IPv6 packet. In most cases latency is measured in milliseconds and you might actually see lower latency since IPv6 servers and routes are currently less utilized. YouTube is certainly faster.
After you have your account and tunnel set up, it is best to set up your tunnel end point on your home router that supports TunnelBroker.net. Anything that runs OpenWRT does this, as do some commercial router firmware. OpenWRT is great for many reasons so you should use that anyways, this is just a bonus.
After your router is able to reach IPv6 hosts (google.com), you would set up and enable a service called radvd (router advertisement daemon) on your router. This will automatically give all the hosts on your LAN instant IPv6 connectivity.
Lastly, you would set up an HTTP ping to Tunnel Broker to let it know if your IPv4 address changes. Once again, OpenWRT does this out of the box.
I am writing this from my phone, so I cannot provide much specific advice, but email me at igor <at< igorpartola.com and I will answer any specific questions you may have.
Since I wrote my original comment, I went out and started doing more research into this. At this point, I have my main home server[0] now accessible via IPv6 using freenet6. However, I did apply for SixXS, so I can hopefully put my entire house on it (my router's an Asus WL-520GU, which currently has DD-WRT on it, but is compatible with OpenWRT). Though I'll be looking more into the Hurricane Electric service.
I'm really liking this, since now I can access my devices directly, without having to try to negotiate firewalls and weird subnets!
Thanks very much for the advice!
Re: DD-WRT, last I checked it did not come with an IPv6 firewall, ip6tables. I ended up recompiling it using what seemed to be Voodoo magic (match the arch, release, kernel version, etc.) OpenWRT comes with the firewall on by default and a web GUI to control it.
As for the different tunnel brokers (of which TunnelBroker.net is one), I should not assume you are in the US. Freenet6 is great. So is SixXS, though watch out to not lose credits and have your tunnel cut off. TunnelBroker.net is run by Hurricane Electric, has been rock solid for me and their support (for this free product) is fantastic.
Also, check out dns.he.net to set up reverse DNS, or just use them as your DNS provider. This service is also very nice and free.
"Running dual stack servers lets you reach more customers."
Is there a single ISP in the world that is IPv6 -ONLY- to the end device?
The main advantage to running v6 on your servers (aside from being a good citizen) is that it allows you to distinguish individual customers for abuse detection, who'd otherwise be NAT'd to the same IPv4 address by their ISP. And ISP NATs are going to become more prevalent, because there just aren't enough IPv4 addresses to go around.
What IPv6 provides is an escape hatch; a path that avoids the NATs and tunnels.