Can we connect the power in our regular home socket or is it necessary to have a 12v battery? Are there any differences?

Connecting to a regular home socket is very dangerous - water will conduct electricity! The 120Vac or 240Vac is high enough to kill you. Since water will allow enough current to flow to be deadly, you don't have to touch the wire that is in the water, just touching the water will be deadly. Of course, there are appliances that are meant to work under water, they have special seals that prevent water from getting inside or they are totally glued shut. Stick with 12v for underwater projects, which a is low enough voltage that will not kill you when it is wet.