When editing e.g. a release I get an overview on the last tab of my actual changes, with green/red colours for added/removed content (like a colour diff more or less).

This is really helpful, but when I change e.g. a title from
"The King's Speech" to "The King's Speech"... the whole words "King['|']s would be coloured,.. not just the changed apostrophes.

I think it would be better if just the actually changed content would be colourised, because then I have to check far less other content (i.e. the surrounding characters) for accidental changes/typos.

And I further think, that even if just the apostrophe would be marked (or just one letter or so)... it would be still visible enough when comparing the two sides.
If you disagree, then a better solution would perhaps be the following:
Only highlight (i.e. background colour) the actually changed content (as I said above)... but add a colourised underline below the whole word.

Obviously the above leaves one thing open:
What if content is only added or removed, e.g.
"The Kings Speech" -> "The King's Speech"
or
"The King's Speeech" -> "The King's Speech"

In that case, I'd propose to add small coloured bar at the place where the addition/removal took place e.g.
"The King|s Speech" -> "The King's Speech"
respectively
"The King's Speeech" -> "The King's Spee|ch"

But the above is just for visualisation what I mean... it would IMHO be especially important that no real character (like the pipe symbol I used above) would be added... not even a special character like a narrow-non-breaking-space or a zero-width-joiner or something like these... because when people copy&past stuff from the diff.. they would copy that crap along.
So that idea only works if there is some way in CSS (or so) to simply add a coloured bar/line/etc. which isn't selectable text.