Basically, P3 aims at aligning the incentives in the right way in the interest of taxpayers. For example, when you ask a consortium that's going to be responsible for designing, building, and maintaining an asset over the long term, they can make decisions that will minimize their costs over the long haul.
When you ask an entrepreneur to build you a road and you award the contract to the lowest bidder, they will not propose something with concrete. It's very expensive and the entrepreneur is not responsible for the long-term maintenance of the asset, so they have no interest in proposing to build the road with concrete.
But if the bidding consortium is responsible for the long-term maintenance of the asset, they may see the economies of scale. The life-cycle costs will be lower than paving the road with a lower-grade asphalt and replacing it multiple times over the life of the asset.