Ash/Swamp Ash has different weights in as much as wood is not uniform. I recently worked with Warmoth on a medium scale build and paid an up-charge for "extra light" swamp ash. So, it doesn't seem that there are any really and guarantees. I know that swamp ash tends to be lighter, but the drift I got is that all wood varies.
I have a P-bass that is ash with maple board, and a medium scale PJ that is extra-light swamp ash with a roasted flame maple board. Both are passive. Both are different from one another because of these characteristics, but it keeps going. I have a standard Fender Mex P bass pickup in the P bass, and a Lindy Fralin PJ set in the medium scale.
Taking all into consideration, I would say that, with EQ flat and all things that can be equal being equal, my P bass sounds a little more earthy and less punchy, and the MS PJ sounds a little more focused and a little more punchy. Both have a very cool vibe. I would say that if I put my mind to it, I could probably eq both to be very similar to one another. If my collection grew past 2 basses, and needed to be reduced, It would for sure only come down to those last 2 and stick there.
There are lots of considerations beyond wood. If you pulled the trigger on one, and regretted not getting the other, I'll bet that tone-wise, you could walk it back a pretty acceptable amount by changing out potentiometers to a different K value, maybe swapping pickups, aluminum bridge vs brass or steel bridge, bone or even a metal nut vs. other materials, string choice... There are things you can do beyond wood.
Me, If I were to build any future bass I was going to get, I'd go with light weight swamp ash and roasted neck every time, and then tweak for purpose via making choices with everything else I just mentioned. Good Luck!!