In Sublime Text 3, how to have shortcuts for "Build and Run" and "Build only" separately like it was in Sublime Text 2?

Addings this to your sublime-keymap should result in the expected behavior:

{ "keys": ["ctrl+b"], "command": "build", "args": { "variant": "" } },
{ "keys": ["ctrl+shift+b"], "command": "build", "args": { "variant": "Run" } },

However you might want to remap keep the list of options to alt+b:

{ "keys": ["alt+b"], "command": "build", "args": { "select": true } },