I'm not of the opinion that "a P is a P is a P". Every bass is going to sound different to some degree, and that doesn't necessarily have have anything to do with different pickups, though it can.
Yeah, the 70's bass is brighter, but it has a lot more "life" and "resonance" than the '16. That's not an issue of pickups, but the wood itself. Maybe it's attributable to the age, or it's just the specific pieces that were used in each bass, but it's not at all surprising to me that they'd sound significantly different. The necks will contribute the most difference to the sound.
How do they sound acoustically, without amplification? In my experience, you should be able to hear and feel a difference in how each bass resonates without any amplification.