To begin with, some basic tasks are faster. Keyboard shortcuts help you navigate around without much mouse click, and a dedicated button can add some smilies or hand claps to your latest tweet. You'll also find an upgraded trend section for times when you want to track the planet's pulses, and search promises to be more sophisticated.
This does not appear to include some of the great updates mentioned below, such as status updates. And now, there's no edit button. Still, it's not that Twitter is giving its web client over and over after focusing so much on its attention on mobile users.
Just when you'll see it is another story. At the moment, it's limited to a portion of the audience through an opt-in button. You'll know if you have access, then. It could be a while before the makeover is broadly available.