I am a Louvre-lover, hahaha so I vote for that! Last October I went 3 times in 3 weeks. I am not schooled in Art but there is so much to see there. I generally pick out a wing or a floor that has a piece I'm interested in and just do that, then leave after 2 hours or so.
Last time my focus was on seeing a piece of art suggested by a fellow forum member, Lyndash which was the Tomb of Philippe Pot.
https://en.wikipedia.org/wiki/Tomb_of_Philippe_Pot
It is SO cool but it took 2 tries to find the room open as it was closed on my first visit (even though the website did not note it was closed)! Previously I've gone on a "hunt" for the Vase of Alienor d'Aquitaine which was her wedding gift to her first husband in 1137.
I generally try to stay away from the Denon Wing where the Mona Lisa is located although this next visit I am going to try to make it to that room to have a good look at the Wedding Feast at Cana which is on the opposite wall from Mona. In fact, she is sort of looking at it, haha!
I also recommend the very readable book by Elaine Sciolino, Adventures in the Louvre. She was a journalist for the NYT posted in Paris.
I have not been but I understand Vaux-le-Vicomte is decorated for Christmas and is wonderful. I've been to Chantilly in a warmer month and enjoyed it thoroughly.
You might consider visiting Basilica Saint-Denis if you haven't been. It is the burial place for many of the Kings and Queens of France including Marie Antoinette and Louis XVI as well as Alienor d'Aquitaine's 1st husband, Louis VII. It's essentially a time-line of history.
In Paris you might consider visiting the Archeological crypt which is below the pavement in front of Notre Dame. They were digging in the 1970's (?) for a parking area and oops....came across medieval and Roman ruins at their feet.
Is there a particular historical era that interests you?
If WWII, then the Museum to the Liberation and Resistance is free, closed Mondays and very interesting.
History of instruments and inventions? Musee des Arts and Metiers
Hunting and wildlife (only if taxidermies heads don't bother you)? Musee de la Chasse et de la Nature.
History of Paris - Carnavalet Museum (free)
I love London as well but especially if you just have 3 days (is that 4 nights?) I'd stay in Paris. Honestly, forum members can fill your time!
editing to add: I see I was busy typing away while Kim was to the point about the Carnavalet!