While other users gave great descriptions on which chip is better, to give an answer on a higher level, the difference comes to the resources available on each chip. As an example, check out the first table from Anandtech's review regarding the 980ti.
The first row is cores, which is the biggest indicator of relative performance across the same architecture. So although the 780ti has around the same number of cores as the 980ti, it is based on the Kepler architecture rather than Maxwell, and cannot be directly compared. When you buy a video card, what you mostly pay for are cores (translates to a bigger chip) and architecture.
Texture units and ROPs help the cores to code data, and so generally vary according to the number of cores available.
The core and memory clocks are also extremely important. They are analogous to CPU and RAM clocks, and generally faster=better.
Memory bus width can be important, but generally just increases bandwidth. Most of the time the buses won't be too narrow on these higher-end chips, but may be an important factor on mid-range cards.
Since this is an extremely brief overview, I'm missing a lot of details, but that's a simplified explanation of what makes one chip better than the other. There are a lot of nuances with bottlenecks, TDPs, power management, etc., but those tend to be chip/board specific.
AMD's chips work exactly the same way. I would look into both camps before making a decision. At this moment, for most cards I would lean towards AMD since they have superior price/performance metrics. If the games you play run better on Nvidia hardware or you miss features as you mentioned, there is nothing wrong with leaning towards Nvidia. At $650, Nvidia wins hands down in my opinion, but make sure that you even need that for 1080p 60Hz. You could probably make it with a 390/970 if you wanted. A 980ti would definitely last a good bit longer. Also Pascal and possibly Polaris (since AMD usually launches a good bit later) are maybe worth waiting for due to a number of factors. Just some things to consider.