They were practically twins as far as the exterior, but very different on the technical end. Britannic was graced with a double-bottom hull and her watertight compartments were extended upward substantially. Of course, there was the addition of lifeboats and the then-experimental gantry davits. Check me on this, but I believe Britannic was also the first Olympic class to use the turbine engine versus the reciprocating engines that Olympic and Titanic had.
As far as furnishings go, there really wasn't any that we know of besides the hospital stuff that I described above. I'm sure there were desks and chairs for writing/reading, and surely tables and chairs for sitting down to eat, but nothing ornate. I believe I read in one of Simon Mills' books that the swimming pool was running and the medical staff would sometimes enjoy taking a refreshing dip when off duty. This is either from a Simon Mills book or Violet Jessop's memoirs.
There was indeed a staircase of some sort where the Grand Staircase would have been, and there was a dome. The staircase, as far as I've been told, was never graced with the ornate woodwork and wrought-iron. There is only one picture that I know of that has Britannic's staircase in it and it's of two or three nurses standing on the landing of the staircase where the clock would go, and you really can't tell much from the photo about the staircase itself.
The dome is gone. Simon Mills made a point of getting someone in to look for the dome because he really hoped it would be found intact but when the divers got in there they found that it had shattered, probably upon impact with the bottom of the channel. You also have to remember that shortly after being launched from Harland&Wolff, the Britannic sat there for quite a while without hardly anything being done to her before she was reassagined to the British Royal Navy as a hospital ship. I doubt the interior of the Britannic looked much different than any other big infirmary. Not much to look at.
Like I say, go through
www.hospitalshipbritannic.com and read through all the info there. It can tell you much, much more than I can. Hope some of this helps.