Think back thirty years. Imagine if ABC or CBS could demand that the New York Times not publish statements made by a politician on the nightly news or aggregate information about the political stances of a representative.
The point isn't to stop deleted tweets from being disseminated at all - that's impossible and I'm sure they're smart enough to know that.
The sole purpose of sites like Politwoops was to undo a feature of Twitter's product. Not only that, but they were using Twitter's own API to accomplish that. Twitter's goal is to set the precedent that that's not ok.
The analogy holds because Twitter is 1:N, not 1:1. They are publishing information to a large amount of people that people use to educate and inform themselves.
And I already stated that doing this for every Twitter user is slimy. But retaining the record of how a politician has communicated with the public? That's in the public interest.