Interesting stuff. The cool thing about Slingshot is that it seems like it'll be relatively easy to bolt it on to existing Rails apps. For Apollo, I think you're pretty much going to have to build your apps for Apollo from the start.
The downside to Slingshot (for some people) is that, since it's rails, you're gonna be sending at least part of your application's source code down to the client to make it work offline. I don't think that's such a big deal, but it could be a dealbreaker for some.
I couldn't find any info on the license for Apollo yet, but I guess it is not Open Source. Neither is Slingshot. It's certainly a very interesting development (especially considering mobile applications), but I would either wait for an open source framework or create my own.
The downside to Slingshot (for some people) is that, since it's rails, you're gonna be sending at least part of your application's source code down to the client to make it work offline. I don't think that's such a big deal, but it could be a dealbreaker for some.
edit: see also: http://www.techcrunch.com/2007/03/23/here-comes-competition-apollo/