Changing pick ups

It is a tone thing different pickups have different areas they work their magic.

 

Different pickups=different tone.

Guitar's and basses are extremely similar, if not the same in how they produce sound so if changing something on a guitar changes the sound, then changing the same thing on a bass will do a similar thing

 

And like guitar pickups some Bass pickups are hotter than others.

 

Good basic question.
The biggest factor in tone is the place where the sound starts, IMO. That's the strings (some would say the hands). Next is the pickups. Amps shouldn't do much more than make it louder. Cab's contribution is better addressed by one of the speaker nerds on this forum.
Pickup changes can result in tone changes from radical to subtle. Basically, if you're happy with your tone, stick with the stock pups and experiment with what your hands contribute - attack, muting, placement relative to neck/bridge, etc. Listen to yourself carefully. Then experiment with different strings. If you still are not satisfied, read a lot about different pups.

I've been really lucky with the pickup mods I've done. They have mostly given me what I wanted. But I read a lot beforehand. You will find a bewildering and conflicting range of opinions regarding modern, vintage, woody, punchy, "fat," "warm," even organic tones. Oy, it's a mess.
Two recommendations I can make are: Dimarzio tend to be very hot, punchy and quiet. Great value for the dollar if you are looking to try something different. I like 'em.
Villex tend to be less hot, very, very richly detailed and musical in an old-school jazz kind of way. Also dead quiet. Expensive, but totally worth it if it's the sound you seek. I love 'em.
Both are passive designs.
Remember, if you try something and hate it, you can always sell it here.
Also, used pups are a good bet, because there are no moving parts, no electronics to crap out.... If there's something wrong, it'll be obvious immediately.

Good luck and have fun!