By the term "Stick ON", it means the heater malfunctions and constantly stays on. Normally heater turns on for a period of time, then switches off again. But when a malfunction occurs it either stays on constant or doesnt switch on at all. Both can have their consequences.
This is pretty much it. The heater has a thermostat in it which is set to a certain temperature, if the temp falls below that then the heater switches on and heats the water up again. Once the water reaches the desired temp then the heater switches off again. The problem comes if the thermostat breaks in the "on" position. Breaking in the "off" position is usually alright, as the water will slowly cool to room temp and then stay there, which is usually manageable by most fish species. However, if the heater is constantly heating the water and not stopping, eventually it will get so hot that the fish will die. The way to get around this is to have a couple of thermometers in your tank and check them daily. Then if you do notice the temp shooting up you can turn the heater off at the plug.
Unless you plan to keep subtropical fish (danios, minnows, some barbs, corydoras, etc etc) or live in a country where the ambient temp is high, then a heater is worthwhile. Thing is, if your room temp sits at the temp you want your tank at for some of the year, the heater will simply not switch on. But you'll still have the reassurance that if the temp does drop (for exmaple during the winter) then your fish will not get cold
As always, it's up to you, but personally, I don't feel comfortable if my tanks are unheated, unless they
only have shrimp/snails in them.
