A bass tuned differently will always sound different. The frequency response of any given string on a bass is modified by the comb filter that the pickup position(s) add, and that filter changes when you re-tune the string - it's a wavelength filter, which... I'll skip the Physics lecture here, just know that you're not crazy thinking it sounds different. It does.
The string tension doesn't change because the strings "aren't used to it", it changes because to resonate at a lower pitch, any given string requires a lower tension. So, you need thicker strings to keep the same tension. Again, you're not imagining things - that's very real.
So, is keeping a bass tuned differently a good idea? In someways yes, but there is one what which gave me fits:
I have almost perfect pitch, and for a while I kept a bass tuned in D standard (with heavier strings) as my "low tuned" option. Sounds simple, keeps the same relative fingerings, but... my brain would hear "C" and attempt to play C where it "should" be - 3rd fret on the A string. But, on a D standard tuned bass, you don't have an A string, you have a G string - C is the 5th fret. After a protracted period of experimentation, it has become clear to me that my brain copes better with a 4/5 string transition (I still have a 4 string bass tuned the same, it just has another string added next to the other 4) than it does with moving up or down the fretboard to find "C". So, now my "low tuned" option is a 5 string. It was a long journey to the obvious solution for me; your results may or may not vary.
The string tension doesn't change because the strings "aren't used to it", it changes because to resonate at a lower pitch, any given string requires a lower tension. So, you need thicker strings to keep the same tension. Again, you're not imagining things - that's very real.
So, is keeping a bass tuned differently a good idea? In someways yes, but there is one what which gave me fits:
I have almost perfect pitch, and for a while I kept a bass tuned in D standard (with heavier strings) as my "low tuned" option. Sounds simple, keeps the same relative fingerings, but... my brain would hear "C" and attempt to play C where it "should" be - 3rd fret on the A string. But, on a D standard tuned bass, you don't have an A string, you have a G string - C is the 5th fret. After a protracted period of experimentation, it has become clear to me that my brain copes better with a 4/5 string transition (I still have a 4 string bass tuned the same, it just has another string added next to the other 4) than it does with moving up or down the fretboard to find "C". So, now my "low tuned" option is a 5 string. It was a long journey to the obvious solution for me; your results may or may not vary.