I also upgraded from the iPhone 6+ to the 6S+ which I received on launch day. I find performance to be substantially better across the board. Everything I do on the phone just feels so much smoother than the prior generation. I have also not had one crash of any app or webpage, which is a huge change from the 6.
It's clear to me that the last generation was pushing the very limits of the 1GB of RAM (hence the app and web page crashing), and the CPU was not able to drive the 1080p display at the smoothness we've come to expect from an iPhone.
This new hardware solves all those problems in a major way. The CPU is faster, the GPU is faster, there's twice the RAM (probably 3x the available RAM to apps), and the interface between the RAM and CPU is far better too (see Anandtech).
When I browse the web on this thing, it feels like using a desktop PC. You can NOT say that about the prior generation.
YouTube comparison videos don't really depict this difference well. It's not about the total number of seconds for a webpage load. It's about the smoothness of the whole experience of the lack of crashing, plus the ability to retain websites in memory.