Although DxOMark isn't perfect, it's a useful metric for comparisons.
By standardising the results of the best cameras in their class (FF, APS-C, m4/3, 1"), you can see that m4/3 is indeed closer to the performance of 1" sensors than it is to APS-C, whilst FF is still a fair bit better than APS-C (with the Nikon's DR being an outlier IMHO). Although not a perfect means of comparison, it does illustrate the point nicely I think. Of course, if we take FF as being 1 stop better than APS-C as conventional wisdom dictates, in the grand scheme of things, 1" isn't that much worse than APS-C for the vast majority of users in the vast majority of conditions.
Obviously this doesn't take into account things like AF performance, price, lens availability, etc. in which case there are significant differences between the various systems.
It's a shame the Nikon 1 family didn't really take off. A really good 1" interchangeable system could really rival m4/3 if done properly I think if Olympus/Panasonic don't produce a significantly better sensor anytime soon.
http://www.photozone.de/Reviews/986-equivalence
This talks a bit about weight and size differences between systems when it comes to the lenses, which is the key point. You can't get around physics. Most of the reduction is in the body, and it's rare for manufacturers to produce a lens that's significantly smaller or lighter (Fuji's 18-55 and Panasonic's 12-32 being nice examples), leading to only marginal reductions overall depending on the combination you're after (telephoto is never going to be pocketable, etc.)