The Sony CX240 is an excellent camera and is just fine for broadcasting stake conferences. Just mount it on a tripod and connect to the Webcasting equipment. If you set it in one position and leave it, the camera angle will probably get boring after an hour or so. For best results have a camera operator control it, but the operator can a distraction to the congregation.
The Sony EVI-D80 is a remote control pan/tilt/zoom camera and is good for static shots, since PTZ movement or recalling a custom preset position is generally jerky. It also requires more to set up. For instance, it doesn't have a built-in monitor so you cannot see what you're doing. The handheld remote control is only good for 15 or so feet, so you would need to be physically close.
The EVI-D80 is best used remotely - with wires that extend the video and control capabilities into another room where a monitor is set up. The result is there no visible camera operator and the camera can even be hidden so most people won't see it. To overcome jerky movements of Pan/Tilt/Zoom action, an additional camera can be added and displayed on screen while the shot of the PTZ camera is framed. Once framed, a video switcher can alternate between the the two cameras...which is also more to set up.
The EVI-D80 has a lot more potential than the CX240 when it comes to remote control capabilities but requires more set up, more equipment and more wiring.