Both are great bikes. IMHO, the FJR wins hands down for reliability, dealer network and cost of ownership. The BMW probably gets the nod for technology and electronic bells and whistles, if that is important to you.
Note that the FJR hasn't been significantly updated in a half dozen years and I believe the end is in sight for this model. Who knows if Yamaha will eventually release a re-imagined FJR sometime in the future...?
I rode the 1250 RT around Europe for 10 days (mostly 2 up) and loved it. We did Stelvio 3 times, and many passes each day and the RT turns quickly and easily, accelerates strong down low, and maybe popped a small wheelie by accident on a switchback.
But I didn't get the quick shifting strategy down at all. I did them all manually or not at all. Also we rarely went over 120 kph so mostly smooth back roads. Smooth suspension on both bikes, but the RT seemed smoother.
My 2013 FJR is faster, more fun above 5k , and just loves to pick a line and stick it! The RT feels lighter and shorter, the FJR feels planted and longer for the sweeps, not the quick little 10mph u turns. Get 2014/2015 if you want elec suspension. I prefer the 5 speed tranny.
I love the FJR in the US, but I can agree the RT might be more at home in the European mountains. Cost of ownership is tires and oil. I added a custom seat too. 55k miles later it's still like new! Try a FJR and point it straight and GO, you'll love it!
I've only ridden my buddy's 2012 R1200RT but comparing it to my 2016 ES brings a smile to my face. The BMW seemed to have a noticeably lower center of gravity so it was less scary trying to maneuver it at very slow speeds. The engine was very torquey but lacks the ultimate oomph of the FJR's power plant. Speaking of torque; you'll notice that the FJR doesn't "twist" the way the BMW does when you blip the throttle. The ergonomics of the FJR seemed more rider oriented. There were a few controls on the BMW that were hard to use while under way. Also BMW's "magic" knob was a source of frustration for me but I suppose you'd eventually get used to that. The ride on the BMW was smoother hands down. At speed however, the FJR is every bit as nimble and loves to carve where the BMW needed to be pushed a bit harder to do any fun twisties. Both bikes vibrate the pegs and grips but the BMW was more noticeable and at a much lower frequency. Ultimately, the big Yamaha's engine felt more refined and smoother. It was much more powerful feeling, especially in "S"port mode. The BMW's tranny shifted more smoothly unless you tried to go too fast then it tended to balk. The Yamaha's tranny was less refined feeling unless you shift it very fast. It seemed to really like going fast. That's why I'm on my second FJR; I too like going fast. She's also dead reliable.