The former would serve as a staging area and playground for new features. Just as the internet does today, it would provide, roughly speaking, no accountability and no security. My guess is that its main audience would be high-school seniors and college-age kids. In addition, you'd have a minority of techies, and — yes — some vile wingnuts.
The "safe for life" internet's network-layer would have baked-in authentication (eg: part of your IP is a user-id). It would have a protocol for notarization (ie: the ability to have a third party vouch for information. eg: the choice to tie your real name to your user-id, or remain pseudonymous). Its "web" markup would be far simpler and more semantic (no per-site styles, no dynamic features, no scripting).
When someone invents a very useful web feature/paradigm on the old internet/web, the new internet's web-standard could add special tags to support it. So, for example, the NewW3C could introduce a set of "store-front" tags with which one could create an entire online-store without any scripting. The NewW3C would include all sorts of functionality, eg: tags to host a Twitch-like site to stream video, with no JS whatsoever.
With this sort of accountability, the alternate internet would finally provide the ability to effectively moderate — bad actors wouldn't easily "respawn" a sockpuppet or bot account, to evade a ban. It would make commerce and data-sharing much safer (via the lack of dynamic features).
The situation today is absurd. What content do we want the web to promote: interesting photos of Japanese food, by a serious developer... or hackneyed ramblings by a bunch of 20-year old trolls? The internet we have today is "wrong" for most people.