I understand all the benefits of HTTP/2, however, are there proven apples to apples comparisons to show it's speed advantage?
Akamai has a page put up for this purpose, but it seems HTTP/2 at best keeps pace and at worst is slower. I tested it in Firefox. In Chrome, HTTP/2 was slightly faster.
First of all let's verify if you are using HTTP/1.1 or HTTP/2.
Open your developer tools in Chrome or Firefox, add the protocol column and reload the page. you should see a number of images loaded from http1.akamai.com and a number from http2.akamai.com:
If you are not seeing HTTP/2 here then something is preventing HTTP/2 from being used (likely a proxy or anti-virus software) which might explain this. Though I see from your screenshot that "Your browser supports HTTP/2!" so I presume you are using HTTP/2 but no harm checking.
What the test does is load 378 images to make up each globe. Loading so many resources is something that HTTP/1 is particularly bad at, and which HTTP/2 is particularly good at - mostly due to HTTP/2 allowing multiplexing.
So if you are using HTTP/2 but are not seeing HTTP/2 being faster then it's likely due to one of these reasons:
Cache-Control: max-age=0, no-cache, no-store
HTTP header on the resource so it shouldn't be cached, but still - proxies have done stranger things! to be honest loading this site in 0.62 seconds in HTTP/1 sounds too fast unless you are sitting in Akamai's data center, so this is what I think is happening. Try another HTTP/2 test (I have my own one on my blog for example).