I have done a few bridge swaps and never noticed a big change in tone as such...
There is a change in the envelope of notes with usually an increase in sustain.
The really big change was on my Rickenbacker 4003 - this developed the common 'tail lift' , plus the equally common 'folding' of the body at the neck pickup route, which made it unplayable due to a very high action (even with saddles at max low and extra deep string grooves).
I swapped it for the Hipshot replacement bridge (brass version to help with neck dive). This is a straight swap using the same screw holes. I flattened the uneven base with emery on a glass plate beforehand and it was a ten minute swap.
The lower Hipshot allowed me to drop the action a lot.
The difference was quite noticeable - ridiculous sustain - notes ceased to have a 'plucked' envelope and became almost constant like a bass pedal.
Result, a massive improvement in playability, not really better 'tone' but certainly a different sound... (a chunk of foam under the strings helped return sustain to something useful).
With 'classic' Jazz and P basses I think the BBOT gives a more traditional look and sound - with modern basses I like the Fender 007-000-5124 American Standard 2007 HMV Bass Bridge for ergonomics and function.
I never got on with the huge Badass style bridges - mostly as they look crap. 😉 YMMV.