Well, I've been possessing a 5712 since 2012 and have been wearing it more than half of the year in average and even much more during these last 12 months. It has totally taken over my other watches. It is a perfect watch for me, in many ways: finishing and crafting work (especially the case and bracelet), the elegance (partly due to its thinness), the smart-casual look, the great movement (accuracy, reliability, finishing...).
I don't know really about the 3712 as I never had one in my hands but only saw one or two in shop windows.
One thing that I prefer much more in the 5712 is the sidecase's curved design compared to the straight one in the 3712 as I think it fits better the bezel's shape and overall round part of the watch (like the bracelet's cushion-shaped small links). That's a major difference for me. It is now to you to see if things work the same for you.
I don't know if both watches have the same thinness but this might be something you want to check. The difference between 3700 and 5711 makes them very different for instance and I don't know if this is the same for the XX12 references.
When I held the 3700, the buckle's construction was different too, making me prefer the recent models.
All this to say that opting for a 3700 is going for a legend, a thinner look, the original construction, all kind of elements that make its charm and make someone want to go for this one instead of any other watch.
If you are not that much into vintage and these creterias, then I think the 5712 is better as in total phase with todays design, the style, the thickness, the material and adjustment improvements etc...
I went for 5712 and I would do it again. But this is me.
I would see more difference maybe between a 3700 and 5711. The thing is that if it was to be my first one (or only one) I would go for a current model. If it was a second Nautilus, then maybe I would add a previous model to my 5712 (thus a 3700 for instance).
About your late crush: I did the same. For 3 years I heard about Nautilus and didn't know what people liked about them. It is when I finally tried a 5711 during a Patek presentation in Paris that everything became clear in around 2 weeks time.
Another lesson, if needed, that every watch has to be seen in the metal to be judged fairly
Cheers and enjoy the trip
Mark