In a nutshell the 4001 was the earlier version of the (current) 4003. It was similar, but the model evolved over the years and at some point the name changed.
4003 is the main version with the shark tooth inlays, bound body and neck. It is also wired for duel outputs, so you can have one pick up out of one amp and the other out of another.
There is also the 4003s which is similar, but does not have the binding, inlays or stereo outputs. This originally was a UK only model and was used by a certain Mr McCartney.
The 4004 is a modern version. It doesn’t have the scratch plate, binding or stereo and has a different bridge, and pickups in slightly different positions. It is a much more minimalist design but is still the same basic outline. These are not as common as the classic 4003 and 4003s.
Recently there has been a 5 string model introduced with a different bridge, but still based around the same outline.