The difference is in how smart the jack can be. First, it's under legal/licensing restriction, which means you have to pay to play (ignore the adapter for now; it's inconvenient and can also be detected as a particular kind of 'unlicensed' device).
Second is the subtler stuff -- like how the jack can now communicate about the music and the device with the phone, so you really could do things like have exclusive Apple music releases only playable on Beats, or (behind the scenes) have a higher quality version play on your Beats than your other headphones, reinforcing the illusion that the new port and devices are "better" than the old analog one (which is impossible). Or have advertising play on radio, except on headphones that have paid a premium to be "ad-free."
It also opens up much more positive possibilities, like the phone knowing the frequency profile of the headphones that are plugged in and automatically using the right equalizer settings to make it sound great. Or, simply wireless devices being able to charge through the port when plugged in. Or having headphones with memory that can keep a few hours of music on them 'synced' by plugging in so you don't even need your iPhone to listen.
So all in all, it's a set of tradeoffs, giving up some freedom and simplicity, for the opportunity of more possibilities in the future. It's difficult to say exactly how it will be used, but the realist point of view is that it's all ending up as the same analog sound in the end; there's only so much that can be revolutionized. Time will tell.
I'm with mbreese in that I still don't see how having an external intermediate between the headphones and the phone enables this anymore than having an internal intermediate between the headphones and the rest of the phone.
That said, this is only true as long as it's still a 3.5mm plug for the headphones. If headphones start directly integrating with the new port, this equation drastically changes.
Second is the subtler stuff -- like how the jack can now communicate about the music and the device with the phone, so you really could do things like have exclusive Apple music releases only playable on Beats, or (behind the scenes) have a higher quality version play on your Beats than your other headphones, reinforcing the illusion that the new port and devices are "better" than the old analog one (which is impossible). Or have advertising play on radio, except on headphones that have paid a premium to be "ad-free."
It also opens up much more positive possibilities, like the phone knowing the frequency profile of the headphones that are plugged in and automatically using the right equalizer settings to make it sound great. Or, simply wireless devices being able to charge through the port when plugged in. Or having headphones with memory that can keep a few hours of music on them 'synced' by plugging in so you don't even need your iPhone to listen.
So all in all, it's a set of tradeoffs, giving up some freedom and simplicity, for the opportunity of more possibilities in the future. It's difficult to say exactly how it will be used, but the realist point of view is that it's all ending up as the same analog sound in the end; there's only so much that can be revolutionized. Time will tell.