Keeping power to the RV will keep the batteries charged, the refrigerator cold if you leave it on, and the heater will keep your pipes from freezing if you leave the cabinets open. If it gets below freezing for any length of time, you will want to winterize the plumbing and not trust to the heater to keep things from freezing up.
New to this winterizing thing...hadn't thought of the plugged in during the winter thing. I live in the Boston area so am a bit scared of the winter and my new to me RV. So if I winterized everything, kept it plugged in, and a small heater going...I'd be in pretty good shape?
There are several advantages to keeping it a bit warmer inside than out during the winter. As someone said, clear the water out, and winterize as you should because.... POWER CAN FAIL and this usually happens in the worst of weather
If you keep it plugged in monitor your batteries. With the Intella Power converter in my rig they do not often require attention, but monitor them anyway