OK went for a walk today on local lane it dosn't have the normal red by-way sign but a black arrow on a white background and the wording public road
Q ..are they one and the same thing?

Q are the black and white ones Old style and the council have just not replaced them

Signposting is hopelessly inconsistent across the country. There's plenty of byways which are not adequately signposted. I know of many which are definitely BOATs but are signed as bridleways. But the vast majority simply don't have signs. Also, there's many private tracks which aren't signposted as private, and look identical to legal byways, so merely looking at the lane and its signs is never good enough to know its proper legal status.

A good starting point is an OS map of the area. For a definitive answer, you'd need to contact the rights of way officer at the local council, etc etc (I think there's a sticky somewhere on here which explains).