The first thing you might use this for is collaborative editing. Braid can give you the power of Google Docs at any HTTP URL, without writing additional code.
This also improves performance. Instead of using heuristics to determine when a cache needs to be reload (cache-control max-age, last-modified, etags), Braid will automatically push all updates to caches -- guaranteed. That means you never need to force-reload a page, or force-clear a cache. Also, updates are sent as minimal diffs, rather than re-sending the entire resource whenever it changes. This saves a lot of bandwidth and a lot of round-trip latency when loading a page.
As a third practical example, Braid makes it very simple to read and write data from multiple web sites. By collapsing time, braid implementations let you write code that manipulates state at any URL as easily as a local variable. It doesn't matter whether state is located on your server, or someone else's server, or distributed on everyone's servers and clients. It's all equally easy to interact with.
This also makes it easy to write a new UI for an existing site.
As for Paxos, yes, CRDTs are an alternative to Paxos, and CRDTs can be used in Braid. CRDTs have some performance improvements over Paxos -- Paxos chooses a leader, and whenever the leader is unreachable, a new leader election takes place which requires a couple network round trips before any new edits can be broadcast. CRDTs are always editable, and edits can always be broadcast.