I think it would depend on the stock bridge and saddles and what you are exchanging them for. The answer is maybe it would change all of that for better or worse.
IMO, the best reason to change a functioning bridge is to improve adjustability. A new bridge might provide a change in tone, but I wouldn’t expect it. Some people will change a bridge just for looks.
Changes playability, tone and feel. Every bass, every time. The people that say “it doesn’t make a difference” make me laugh. Everything makes a difference. The question is, how much does that difference matter to you and do you like the change? Literally every variable has an effect on a musical instrument’s sound or feel.
It CAN affect all of these. How much and how noticeable it is largely depends on the components already on your bass. A lot of the change that you'll experience will be in aesthetics, durability, and feel unless you have super hearing or are comparing with computers. For example, changing the stock bridge on your Fender bass for a similar bridge made by Gotoh probably won't change what you hear much. If you change from a vintage style bridge to a high mass bridge you're going to hear some differences. Is it a magical, tone shaping miracle? Nope. Is it worth it? Depends on your preference/budget. Changing a crappy plastic nut to tusq or bone will probably improve tuning, intonation, and - in a smaller way - tone. The biggest advantage to me though is that these upgraded materials will last a lifetime while cheap plastic gets worn down quickly and causes all sorts of problems.
I answer that question with another question: what problem are you trying to solve? IMHO there is no reason to change a bridge (or other hardware) unless there is a specific problem or shortcoming that you are trying to solve. Otherwise, you're just throwing hardware at an undefined problem hoping to see an improvement - but you may not even be able to define what improvement you are trying to achieve. It's not a lot of money for a bridge or tuners, but I suggest that any modification you make to an instrument be something you can reverse with no damage or visible change if you decide the modification was a bad idea. Then you preserve your options.
Well... - Some people swear it will effect the tone (or sustain). Other people swear it doesn't. Up to you, I guess. I do know that the Babicz Full Contact bridge on my new Gibson T-Bird does sound slightly different than the 3-point one on my '13 T-Bird. But, IMO it's nothing that I'd replace the 3-pointer for... - Yes, it can definitely effect the string height; to the point that you might have to rout a place for it to sit in, to work properly. I'm dealing with that situation right now, with an old Kramer Forum III. Sometimes you can shim the neck to compensate, but the Kramer's a neck thorough... - Tension? Honestly, I can't see how. That's purely a function of string length and gauge. Other than that, about the only other thing I can think of, would be string spacing, and whether or not it fits well on the body - as in, not hanging off the body's ass end, so that the intonation distance will be right. But, other than replacing a bridge that's just so atrocious that you can't live with it another minute? Except for making you happy, I'd have to say; no, there's no really substantial benefit...
Like others have said, improved adjustment capabilities, but otherwise, unless the original bridge is broken, I would say no.
Depends on the level of crappiness of your stock bridge. Any good quality instrument rarely needs a bridge replacement..
Some people change stuff just for the hell of it (looks in mirror) I fitted a hardtail bridge. to my heavily modified P bass. (Started off in life as a Squier) then converted it to string through body. Lot of point changing the bridge. But I needed black hardware Black on black on black. Laurence
Himass bridge vs fender bend bridge = Snappy tone vs woody tone. If you want snappy tone on traditional bend bridge, try Wilkinson brass saddle bridge. wilkinson brass saddles bass bridge | eBay
Any change in tone will be tiny to nonexistent in my experience, but it’s nice to have features like adjustable string spacing, easier intonation or action adjustment, etc. A heavier bridge can counteract neck dive, or make it body heavy, depending on balance. Expecting some magical improvement in tone or sustain with a bridge is guaranteed disappointment.