For me, the best reasons for having a version of Star Trek that connects to the what's been seen in the past, yet can be different or similar to varying degrees is best imagined when I compare the old and new Battlestar Galactica shows side by side.
The first example that presents itself is a personal favorite storyline of mine in the old series, the visit to Kobol, where the characters visit a mysterious, ancient homeworld that their legends tell them they came from. They do some cook archeology things, there's fun action, and a hard-hitting change occurs for a couple of characters on a deeply personal level. It all goes down in about 1 or 1 and a half episodes, and was well remembered years after I had seen it as a little kid.
The core story was taken by the new series, and expanded and adapted to fit the sensibilities of the new show's style. Some of the consequences of the new show's creator's decisions in adapting the story expanded the scope of what was happening for the characters and story, and the original core story idea inspired material to last for something like 7 or perhaps 9 episodes, forming a huge epic built on that much smaller but still epic idea of the visit to Kobol. Breathtaking.
This big storyline in the new series started as a big time season ending cliffhanger for the new series, after utilizing the style of the new series to cultivating many episode built on entirely brand-new story ideas.
So, the new version of BSG early on maximized it's potential for new stories, and took a small but significantly memorable story idea from the old series and got an extraordinary piece of television drama out of it.
We now have a totally different perspective on how the TOS characters relate to each other. And we have the potential for totally brand new stories. And we have the potential for an old favorite to work in a totally unexpected way.
I know people argue about Khan. But it's a well known example. Strong in the minds of a casual viewer. We could see a version of Khan who has already been set loose on the galaxy. Not one guy with a handful of followers trying to hijack a ship, or a vengeful madman who captures a ship to battle one-on-one, ship versus ship. What about seeing Khan as a prince once again, with political power in the 23rd century? And dreams of continued imperial expansion and conquest. A formidable opponent at many different levels, politically, tactically, or physically.