Public law deals with the relationship between the individual and government, whether local or national.
Private law deals with the relationship between individuals.
Planning law is concerned with the control of development.
Land law is concerned with the rights that individuals have in land.
Planning law and land law operate without regard to each other.
Planning permission is required for many types of development that involve building (including alterations) or changing the use of land. If you obtain planning permission it means that the planning authority are satisfied that you have met certain criteria. It is a common misconception that if you have planning permission you can carry it out willy-nilly. Planning permision should be thought of as simply confirmation that the planning authority has no objection to what you want to do. Planning authorites are not concerned with private rights - this would make the planning process too complicated - they are only concerned with planning considerations.
If you have planning permission to build over a right of way, this does not give you permission to obstruct the way.
If you have planning permission to erect a building that will obstruct the light to a window that has a right of light, this does not give you permission to cut off the light to the window.
If you obtain planning permission that can only be implemented by going onto your neighbour's land this does not give you permission to go on your neighbour's land.
If you have planning permission to carry out works that need a Party Wall Etc. Act notice to be served, the notice must be served.
If you get planning permision to build on land that is subject to an enforceable restrictive covenant against building, this does not waive the covenant.
If you obtain planning permission to change the use of your land you must still comply with any enforceable covenants relating to use that affect the land.
If you obtain planning permission to build on your neighbour's land this does not give you permission to build on your neighbour's land.