Think of connections as the virtual representation of the physical wires that run to and from your system.
There is an HDMI cable between your Roku (or cable box or whatever) and your AVR, and another HDMI cable between your AVR and your TV. Both of these would be configured in Control4's connections - the HDMI out from your Roku would be connected to the input on the AVR, and the HDMI out from the AVR would be connected to the input on the TV.
With lighting, connections allow installers to "bind" a light switch or dimmer to more loads than the locally wired load, or to bind keypads, etc. that don't even have a load wired to them to control lights. The lighting scenes created in the Advanced Lighting Agent also expose a connection point, in the same vein as the HDMI on the back of the TV. Likewise with the light switches and keypads. So if I use the connections tab to link the output of a keypad button with the input of an advanced lighting scene, that button on the keypad now controls the lights.
The connections/binding method is generally viewed as slightly more reliable, quicker to activate/deactivate, and more likely to "just work". However, as you know, the downside is that they can't be modified with Composer HE. You can change the contents of the scene, but you can't remove or add the scene from a button. The same end effect can be done with programming: "when the button is tapped, then active scene x." This takes slightly more processing power however, and in a very busy system can seem very slightly laggy.