This is my take on it. Every instrument has certain sweet spots on it. On my upright the low G, Low A on the E string, the F on the D string, the Bb on both the A and G strings, and Eb on the A string all have a phenomenal growl to them. It really "grinds" there against the keyboard. So, as I've gained experience with the instrument, I know to really milk those notes when they pop up in my bass lines.
If I have a choice of which "A" to use, I'll go with the A on the low E string if I can because of the grind there. Also, there are certain notes where the way I place my fingertip pads (on an angle) on the keyboard creates a particular tone. It's only on certain notes, and again, I know to milk those notes. It sort of comes unconsciously after you've bonded with the instrument for a while. Along with this is vibrato. I know where to do it on various notes for maximum effect, and how to sort of bend the strings on certain notes to create a really unique tone as i play a run.
Last of all -- attack and release of notes creates a fluidity in bass lines. I often have to take the written notes on some songs, or patterns I create myself, and then figure out the best way to hammer on and pull off various notes to give me time to get from one note to another or embed "groove" into the bassline. Also, to create a flow in the bassline so it doesn't sound stumbly or introduce unwanted string noise. I think this influences the other aspects of tone, to create a unique sound in your fingers.
I think it's one reason musicians love to stay with a particular instrument. Jaco's fender jazz, Marcus Millers Blonde jazz, Sheehan's double precision pickup.
Complementing this is amplification equipment. After a while you get to know the kind of tone you want that amplifies or complements the natural tone on the instrument and its sweet spots. So you know the balance of treble, bass and mids that you need. I can get a similar tone on a wide variety of amps and cabs. The biggest impediment is how complicated the manufacturer has made the controls. I have a Peavey Pro Bass 500 and it has such detailed controls I have a hard time finding the tone I want. Perhaps that's why they don't make them anymore. See below:
Peavey Pro Bass 500 - Yahoo Image Search Results
But give me a relatively simple bass amplifier, I can make it sound the way I want. The room you are in influences the tone as well, and some rooms are just boomy. But in decent acoustics, the tone from the amp for me, is driven more by the complexity of the eq and gain controls on the instrument, not the amplifier itself. Complex controls make it harder to find good tone fast.
