I see many responses speaking to tone. Yes, certainly the strings affect the tone.
What about playability? The feel of the instrument. If an instrument just isn't comfortable to play, has anyone found that changing strings to have a profound improvement on playability? This is assuming a proper set up, of course.
I have a touch that varies from subtle to extremely heavy depending on genre and mood, and I have hands that can kill strings. When I was young, I had a lot more oil in my fingers when sweating and would kill a new set of strings dead in a week. I even broke strings back then (including a low B during one memorable session!) . Now my strings last much longer and I don't break them, but they will still go dead after a while from normal use.
This is where not just the tone but the FEEL is affected. Dead strings cause more friction. If the dead strings have funk on them, it's even worse. Does this affect my (subjective) opinion of a bass?
Absolutely. Unfortunately.
Example: I have a bass that I'm in love with for the past year or so since I bought it. This is bad news for a few of my 10 basses that are lower on the totem pole of affection. If I haven't picked them up to practice on enough, the strings have oxidized, which feels horrible. The psychology of this is that it reflects poorly on that instrument, which is unfair. The instinct goes like this: "I haven't played my SoundGear in a while... [picks up, plugs in, tries to play] "UGH! This is horrible!" I know it's because the poor thing was neglected, and now I will neglect it even more if I can't be bothered to spend the extra money to change out the strings.
My solution to this problem is to use the coated strings from DR like the Red Devils and Black Beauties for my instruments that I don't gig on regularly. These strings have a slightly muted tone compared to my regular strings, and they have a unique feel, but they stay nice and snappy for months longer.
The bottom line is that clearly the kind of strings we use matter. If we can notice dramatic differences between different types of strings (roundwound vs flats) and if the age of strings makes a difference, it stands to reason that more subtle differences will affect the playability of a bass as well.