As the sibling points out, if both site and device support v6 the intermediate translation isn't required.
It'll be NAT64.
It means that the phone doesn't have to talk through a network-translation box, which may add latency. For some things (gaming?) this may be important, while for others it may not be.
Further, the price of each IPv4 address is going up, and so having your hosts on IPv6 may allow for the lowering of your costs, which you can pass onto your customers or add to your margin.
About a year ago they were US$ 30/IPv4, and they now seem to be in the $50 range:
Your device's network stack first learns if your mobile ISP supports ipv6 or not, and based on that and DNS reply decides how to connect to a given website.
(Most US carriers have IPv6 enabled and issue IPv6 addresses to their phones.)