Its a big processor/spec upgrade
The iPhone 6 was essentially an iPhone 5s in terms of hardware and processing power, notably chip performance and RAM, plus NFC and a larger screen... Its not a stretch to compare the A7 & A8, viewing the two chips in similar light, given their capabilities, and further the two phones used the exact same type and quantity of RAM. The cameras were very similar as well.
The 6S camera is improved from the 6, imo, but as others stated its not as noticeable a difference as say the 4s was over the 4, or the 4 was over the 3gs.
The main upgrades come in features and processing power:
- 3D touch isn't going to be a big deal for everyone, and some will use it more than others, however speaking for myself I find it an excellent interface and use it daily. Its now how I access the app switcher and it makes multitasking so much smoother and "flow" better over all. Also, I am getting the hand of "peek and pop" and when using an older iPhone, I wish I was able to do that. It becomes natural and it is much less of a gimmick than I thought it would be.
- The "second gen" touch ID is super fast. I love it.
- The new taptic motor is how phones should vibrate. It feels great and works like a true, silent vibrate - not like the small motor which is still audible to everyone.
- Live photos is nice, and neat, and I use it, but its not a "must have" feature. It looks cool when you have it as your lock screen wallpaper and activate it via 3d touch.
- The processor is the first big leap in CPU and GPU power since 2013. The 6s is powerful, the benchmarks are crazy good and it reflects in how it feels - fast, though in every day usage for general items like searching emails and sending texts, making phone calls etc.. it isn't significantly faster on iOS 9 from the 6 or 5s. They all run iOS 9 well, and to the average user the speed increase is incremental. That said, from the perspective of raw power and future proofing, the processor should keep tempo with later updates, which is big if you plan to keep the phone for a few years.
- 2GB of RAM is the big one. Had the 6 come with 2GB of RAM, I probably wouldn't have purchased a 6S. This is the biggest difference in terms of day-to-day use, since pages stay loaded, apps stay loaded, and there is no app refreshing or anything of the sort I have come across yet. On my 6, I couldn't have more than 1 other app running plus 3 Safari tabs without any refreshing - this was a pain with multitasking, editing documents, filling out forms etc... with the 6s, I have not yet run into a single app refresh or single safari tab reload. Most I have tested are about 12 Safari tabs and 5 other apps, 4 of which were Microsoft related (powerpoint, excel, one note, outlook) and the 5th being safari. That is a huge difference.
These were probably the most notable things for me, and made the upgrade worth it. It all depends on what you use your phone for - for me, it is a full tablet replacement and I do much of my work on the go so I need something that can provide me with this level of workflow. When I used my 4s over the summer, I hauled my laptop around too and used my phone more for social media/messaging/phone calls.
Now I can keep my laptop at home or in my bag most of the time, only using it when needed or sitting down somewhere comfortably.
Not everyone will need a phone for that kind of usage though, and if you aren't crazy about multitasking and are happy with the 6, the 6 is a fantastic phone and very powerful today. It still takes some of the best pictures on a smartphone and, in my opinion, has a fantastic & comfortable design.