Jason brought in two of his Monarch 4's for me to try because they were very similar to the setup I was envisioning (hope I remember his specs correctly!). The first that I tried had an Ash body with a Walnut tone block, and the second was solid Alder with a dovetail neck joint. I believe both had Braz RW FBs.
The first thing I noticed was - "Damn, these brand new roundwound strings are WAY more alive than those dead flatwounds I've been playing!". Jason also noticed that I have a heavier touch than he does, as I was getting some fret buzz from right hand plucking. We adjusted the specs on my to-be-built bass to have a medium action (instead of low) as a result.
As far as the tone between the two basses, with a fairly neutral preamp setting (I also tried them passive), the main difference I heard was in the low and high registers. On the Alder bass you had this creamy sort of throaty growl right around the low F and G that sounded really amazing to my ear. On the Ash body bass, instead of that rounder growl, you had a slightly more defined sound.
I'm a brass player, so for me I would describe the low end on the Alder as having a more tenuto sound with a softer front end, while the Ash had a more marcato sound with a harder attack.
To my ear, the upper register on the solid Alder bass maintained the smooth character of the low end, and was really pleasing to listen to, though it might get lost in the mix without some preamp magic. The Ash had a very clear high end that definitely would cut through.
It might have been my untrained ear (or the Braz RW FB on both basses evening things out) but I couldn't perceive any noticeable difference in the mids between the instruments. Both just sounded great.
In the end we decided Alder is for me - I'm not playing any blazing jazz solos in the upper register that need to be at the front of the mix, and having played Alder my whole life it really is the sound I know and love. I'm throwing an Ash tone block in mine and we chose a very bright fingerboard, so I'm imagining the balance on the final product will be great!
(I just realized I capitalized Alder and Ash throughout this post - I think I might be obsessed with wood...)