Back in the day (for me, 1972)...about the only Electric Bass book available was the
Mel Bay Bass Method. Somewhere in there, it sez, "...the advanced player will not use open strings...the reasons are obvious". Or something to that effect.
There was no explanation, though. So, for years, I just ran with that.
Eventually, though, I realized open strings could be used...and should be used in certain circumstances. A great example is the walking bass mentioned above by floppybass.
I used to kill myself trying to play the bass breakdown from "All Right Now" without using the open "A"-string.
Another: Try playing "Liberty City" without open strings.
For a chromatic climb A-A#-B-C... back in the day (by the gospel according to Mel Bay), I woulda played a fretted "A" on the E-string & then shifted down to the "A"-string for the A#-B-C...an open "A" works great in lieu of "the shift".
Anyway, yes...an Open "A" & vs. a 5th fret/E-string "A" sound different...but so does a an E-string/8th fret "C" & an A-string/3rd fret "C", etc.