Going back a while, my TA sub-unit had 4 or 5 from each regiment.
They would speak of differences in ethos between their 3 battalions (I said it was a while, at least 2 had been on the funeral & coronations in 52/53).
Overall they agreed that the Grenadiers could be a bit 'Regimental', even in social settings, while the Coldstream were more relaxed and comfortable with themselves.
From my more limited interactions with the two (while serving and since), that's about right.
In pure soldiering and role, I'm well out of date.