Ok, that is odd: there's no difference between the data connection used by Snapchat or Spotify and that used for web browsing, so one working and the other not is weird. The ping test results seem abnormal to me, but shouldn't prevent you using data for web browsing: my cellular connection passes ipv4 and ipv6, but the WiFi on the train I'm using to send this (from a web browser) also fails ipv6 (ipv6 is the new version of the internet protocol, but ipv4 is still universally supported and I'd not expect many websites to rely on the extended features of ipv6). Of course diagnostic messages don't always tell the full story, but just at face value this seems odd.
In that same menu what does it say for "preferred network type?". There are some options there that if misconfigured could cause problems (though the basic problem of some things working and some not is still a puzzle to me).
A final thought: have you checked it's the same in different locations? It would be unlucky if there were a problem with your home cell rather than the phone, but I've known such things happen.