vs. Château de Fontainebleau

Been to both. 3 times to Versailles. Once was enough for Fontainebleau.

We love them both. Versailles is more about Louis XIV and Marie Antoinette. Fascinating history there.
Fontainebleau is more about Napoleon and it’s fantastic.
We also love Vaux-le-Vicomte, another day trip and amazing chateau.
You can’t go wrong with any of them.
Whichever you choose, read up on the history and you’ll enjoy it all the more.

Versailles is really singular; yeah it is crowded and a bit of a misery. But it is singular. The others are just chateaux -- all worth doing -- I think Vaux le Vicomte is my favorite, but do Versailles once.
We have done half a dozen in the Loire, Sully, Commarque, Sceaux, Ecouen, Malmaison, Chantilly, Vaux le Vicomte, Rambouillet, Vincennes, Maintenon, Fontainebleau and a few others here and there and Versailles is still amazing.

If you want to see a château with the only purpose to impress and for me to intimidate too Versailles is the place to go. You need several visits to see everything (as far as possible) without getting exhausted, for the main building get in early or the last few hours before closing time to enjoy for instance the hall of mirrors without or lesser crowds. Buy tickets in advance online, but retail shops like FNAC selles them too to minimize the time staying in line at the entrance.

From a historical point of view I won’t let them compete, Fontainebleau covers a longer period of French history and for me in this case as interesting as Versailles. It is also more manageable, with one visit you will have the idea having a complete experience, nevertheless another visit or even more can be worth too. The same for Vaux-le-Vicomte, together with the gardens it forms a beautifull combination, even not royal it has a comparable allure and is certainly worth to see. As Susan alraidy says you can’t go wrong and with the smaller châteaux you don’t have to worry about crowds and staying long in line to get in, however Vaux-le-Vicomte needs of the three the most time to get to.

I’m in agreement with the previous assessments. It’s a bit apples and oranges to me, hard to compare, both are worth a visit. I enjoyed the spectacle of Versailles (Hall of Mirrors, I’m talking to you) and truly enjoyed the peaceful, fairytale vibe of Fontainebleau.

OP, you don’t say if this is your first visit to Paris. There are so many wonderful things to discover in the city, if this is your first visit, and you only have 3 touring days, unless you are truly compelled to see a chateau, I might skip the day trip.

We did Fontainebleau and Vaux-le-Vicomte in one [long but fun] day in 2015. Fontainebleau was very interesting from a French History perspective but is a very hodge-podgy chateau from a "beauty" standpoint. There is an excellent if sometimes over winded audio guide. I liked Fontainbleau but don't feel a need to go back. It's a half day chateau at best, though you could opt to explore the town afterwards. Vaux, on the other hand, I could have spent days in. It is just that gosh darn beautiful and also with a really interesting history, as it's effectively the reason Versailles was built. The gardens are gorgeous. We did the dinner and fireworks and found it to be very worth it.

So to answer your question, I would personally then vote for Versailles. It's beautiful and if they still do it, I'd highly recommend the Private Apartments tour. Helps you by not having to wait in line with the mobs and it's also a really interesting tour. You get to see some beautiful objects as well as the theater and a bit better view of the chapel. The main tour rooms are of course beautiful, just expectedly crowded. We were there on a rainy day so didn't get to enjoy the gardens as much but they were beautiful. If you can, definitely try to make it to the Trianons and Hamlet in the back. The Petite Trianon and Hamlets were probably the highlight of the day. The Hamlet is like walking into a fairytale. A long day to see everything but again, worth it in our eyes. I would love to go back someday.

Vaux le Vicomte is amazing -- We were there a whole day and we needed about one more hour -- the size of the garden is totally underestimated until you actually walk its length and it is designed with interesting forced perspective:
https://janettravels.wordpress.com/2010/09/20/vaux-le-vicomte-our-last-chateau/
When I wrote that, I vowed it would be our last chateau but then we did a day trip from Paris to Maintenon a year ago -- and are glad we did.