Rosenborg vs. Christiansborg Castles

It depends on your own interests as they are two very different places.
Christiansborg Palace is the seat of parliament and thus a tour there offers everything associated with that. It is fairly new, being completed in 1928, but it has been the site of power for more than a thousand years. There is another tour going through the basement level where you can see the ruins of the original castle built on the sight, as well as exhibitions of the 4 other castles and palaces that has been built on the site.
Rosenborg Castle is a bit older, from 1608 and smaller than Christiansborg. It was built as the kings country summerhouse, just outside the old city walls. Today it houses the crown jewels.
There is no reason that you can't see bot places from the outside and tjust do a tour of one of them.

I wonder why you didn't include Amalienborg?

At least for a drive through.

Thank you to both of you. The reason I am not going to Amalienborg is that I don't have a car and very limited time.

I think you have misunderstod something. You don't need a car to visit Amalienborg, it is within the city center, just as Christiansborg and Rosenborg. In fact hey are all within walking distance of each other and if you don't want to bother with going inside and are fine with just seeing them from the outside, you could easily see them all in a day.