There are a few things. But the most prominent is the practically infinite number of available public IP addresses which, stated negatively, means you don’t have the problem of address exhaustion or the need to hide behind NAT.
Explained pictorially, here are two IPv6 networks with hosts A, B, C, etc. and routers R1 and R2:
With IPv6, each host can easily (and cheaply) have a public IP, so in you just assign the public IP to each hostname in DNS. Then, if host A wants a Web resource from host F, simple call wget F and you are done. Likewise, host E can simply ping host B to see if B is up.
With IPv4, it is more like so:
Because public IP addresses are so darn expensive, you have to hide the two networks behind NAT at R1 and R2, so that the various hosts can only see R1 and R2. Consequently, if host F and host G both want to be able to provide Web resources, you have to program R2 to forward the traffic through based on non-standard port numbers. So, if A wants a a Web resource from F, it must request that from R2 instead, and if D wants a Web resource from G, then it must request that from R2, and also remember to use a different port number you both agreed on before-hand.
Also, it is pointless for E to even try pinging B because ICMP echo requests don’t use port numbers, and so you can’t tell the difference between R1 and B.
Of course, a stereotypical “consumer” Internet user is not expected to be providing resources out of his or her network. But there is no reason we all have to be just “consumers”. Why not have various computers on your network providing as many services as you would like? E.g., a $40 libreCMC box could be an a chat server, or a P2P tracker, or a dozen other things. Turn-key NAS servers are making file sharing, media streaming, and such like easier than ever. Run a game server. Check your favorite IOT devices (refrigerator, toaster?) at home from work, directly over IPv6, instead of having to work through some 3rd party “cloud” service. Your imagination is the limit.
The (less knowledgeable) security folks will say “well, my network is more secure if hidden away behind a IPv4 NAT”. Nonsense! It is just as easy to tell an IPv6 firewall not to let any traffic through, as it is setup and maintain an IPv4 NAT.
So, if you aren’t already on the IPv6 Internet, flip on 6in4 in your libreCMC router, and start having fun!