When Rickenbacker introduced the 4003 bass the 4001 was still in production, in fact it lasted until 1985 and with relatively little between the two visually and general sound characteristics, its very existence may seem curious.
The truth is, the 4001 had a few design problems - particularly with the truss rod adjustment cavity at the headstock. As the neck was very slim there was actually very little wood around this cavity, headstock breaks were always a danger (personal experience here) and neck bowing could be a problem.
The 4003 was designed to take care of that by adding extra meat to the whole neck. Of course this gave it a rather more substantial feel, so the new designation was justified. Rickenbacker also improved the dual truss rod system and altered the electronics by removing the bridge pickup capacitor and improved 'hotter' pickups, which are much better balanced than on the 4001.
Later versions of the 4003 would include a push/pull pot to engage this replaced capacitor. This bass masters the iconic looks and sound that Rickenbacker basses are known for, particularly when engaging Rick-O-Sound, which provided a stereo output jack so that you can run each pickup into a designated amplifier. But whether mono or stereo this will grunt and growl like no other, although as McCartney proved it can also sound very sweet. Also available as the 4003S unbound mono version, 4003FL fretless, 4003S/5 unbound 5-string and 4003S