A slightly thicker and heavier display would seem to give credibility to the recent rumor of a slightly reduced battery capacity. The leaked chipset showed a 10% larger surface area for the A9 despite it going to a 14nm process. This would suggest that many components are integrated into the A9 and therefore more power efficient. Combine that with the energy efficient software improvements in iOS 9 and you have battery life that is probably better than the last generation.
As for speed improvements, it would be difficult to make significant gains when dealing with a smaller battery, so I only expect some moderate gains at best. I think that's completely fine because the processor in the iPhone is certainly fast enough for phone tasks. I expect performance to increase the most when it comes to the graphics chips. I think the increase in A9 die size suggests an increase in the number of graphics cores. There may be an extra processor core like in the A8X but I'm not sure if they can do it on this energy budget. I think the single biggest thing that will contribute to increased performance on the next iPhone is 2GB of RAM.
Another thing that may help is faster flash memory, but I don't think there are any concrete rumors about that at the moment. NAND speed in phones usually lags behind significantly so some improvements there would be great. But that might also depend on the power budget.
As for speed improvements, it would be difficult to make significant gains when dealing with a smaller battery, so I only expect some moderate gains at best. I think that's completely fine because the processor in the iPhone is certainly fast enough for phone tasks. I expect performance to increase the most when it comes to the graphics chips. I think the increase in A9 die size suggests an increase in the number of graphics cores. There may be an extra processor core like in the A8X but I'm not sure if they can do it on this energy budget. I think the single biggest thing that will contribute to increased performance on the next iPhone is 2GB of RAM.
Another thing that may help is faster flash memory, but I don't think there are any concrete rumors about that at the moment. NAND speed in phones usually lags behind significantly so some improvements there would be great. But that might also depend on the power budget.