Connected Standby just leaves your device connected to devices while in standby. So when you turn it on, it is still connected to the wifi network/wired network and/or bluetooth devices. This allows you to turn it on and start using it right away.
The two methods you mentioned are about the methods you would typically use on a laptop or desktop to still continue receiving notifications from certain programs. You turn off Sleep, and your device is always on ready to let you know when something comes in or is happening.
I do not use many Metro Apps, so I do not know how they effect the Connected Standby differently, but I would not imagine they really would. I have some scripts setup in AutoHotKey to keep my unit awake when I need it to not go into sleep mode. And some others to quickly enable or disable sleep mode.
I have not messed around with looking into if the SP3 has 'wake' events. For my desktop PC, I have it wake up from sleep if there is LAN activity. I use this for my remote desktop software.