On iOS in particular it has been proven when it comes to web apps, from a performance perspective they can never truly be as fast or faster than a native app due to the security restrictions around JIT.

I just tried it out and Fastmail have done a good job, probably as good as you can get with a web based interface. In terms of speed, I think they are referring to user interface response and page load times, not actually faster than native speeds because due to limitations in regards to how pages are loaded, although the title does seem to deliberately suggest otherwise.

I would love to see some benchmarks and tests as to how fast the new Fastmail is though.