Chambers Dictionary's entry for 'pigeon' reads..."Originally, a young dove, a dove, any bird of the dove family."
For 'dove' it reads..."A pigeon."
So, basically, there is no difference.
A pigeon was a feral Rock Dove but has now has been renamed, by a variety of twitching groups, the rock pigeon. All doves and pigeons are in the order Columbiformes. If you are from the UK the most common dove you will see is the eurasian collared dove, it one of the great colonisers of the avian world arriving in the UK in 1953.
A white dove is just a pigeon with no colour. It's just that they look prettier!!! Sometimes in certain sunlight conditions you can still see the fluorescent green sheen seen on pigeons on some white doves.