I've gone from the RMHD to the S6. There's a couple things to mention...
1st. I had my RMHD bone stock, without upgrades, because the newer apps (Maps is probably the biggest culprit) were crashing and draining the battery. Bone stock, I used the phone all day and still had battery the next. The S6 will not operate that way.
2nd. I had taken the S6 off the charger Saturday morning, not returning it to a charger for 1 day 16 hours, with 30% battery remaining. I mention that just to counter the stories of horror that the phone will die in your pocket if it doesn't make it to a charger in 8 hours.
3rd. In real-world business use yesterday I had just over 1 hour talk time, 1 hour Navigation and 1 hour screen-on time which ran the battery to 44% by mid-afternoon. At that point, I set it on the Powerbot sitting on my desk. Had I stayed on the road longer, I likely would have plugged into power in the vehicle. With the RMHD, what I did yesterday would probably have left me with closer to 60% battery. Point is, there is a difference.
4th. RMHD was on Verizon. With the RMHD we have simultaneous Voice and Data no matter where we are. With a new phone, you only have simultaneous voice/data if your data is on wifi or if your voice is HD Voice on 4GLTE. There is a sticky at the top of this forum that will explain it a bit more. I've made the switch to AT&T for the purpose of maintaining simultaneous v/d no matter where I am. This may or may not be an issue for you, just something to keep in mind since you're leaving a phone that was about as good as was ever made for Verizon, in that v/d worked throughout all the 3g/4g areas of their network. I believe a Note model or two did the same.
5th. (which might be a repeat of 1 but restated with the other variables mentioned) Had I updated the RMHD in the manner the S6 is, installing Facebook and Twitter and all available updates, the performance would be below the S6. The last time I used it fully updated, I failed to make it to the end of the day. So, the issue isn't just battery, but it's battery, stability and performance.