While it's true to some extent that no matter what instrument you play, you will always sound like you (with regards to character, phrasing or whatever), on a crappy bass (or even a good bass with crappy strings) you will never sound as good as you will on a decent quality, well strung instrument. All nuance and subtlety will be lost - a poor instrument will simply not be able to capture all the details that your fingers are (hopefully) producing. Of course this may depend entirely on what style of music you are playing - if you are plodding away on root notes to some three chord fuzz tone dirge, then there probably ain't much room for subtlety/nuance to begin with, so it undoubtedly matters a whole lot less...
I suspect this also applies for point 2, i.e. audiences for different types of music will have different levels of awareness as to how good the overall sound of the band (including the bass) is, and will also give/not give a solitary flying fornicate to wildly varying degrees...