Advantages of a 'proper' birthing pool over a paddling pool:
Less likely to leak (even the inflatable ones). Sides can take weight, so that you can lean on them or even sit on them. They have handles to grip during a contraction, if you need to. Padded bottom. Probably better insulated (very important to keep the water within a degree or two of body temp, especially if the baby may be born in the water).
We get our BPIAB out a few times every year for a mega-paddling pool in summer/hot-tub in winter.
the sides are MUCH higher. so it is all nice and cosy and it's the right height to lean on which is good. also means that the water level is high enough to 1.) immerse you so you benefit more from the lovely warm water. 2.) ensures that baby stays underwater until whole body is delivered. 3.) means you can have lovely cuddles with you and baby kept warm by the water until the cord is cut.
plus everything that prettycandles said.
i think a paddling pool would be far too flimsy, but most of all too shallow