The A8 chip uses a 20-nanometer manufacturing process, which makes it 17% smaller than the A7, but has almost double the number of transistors. The A7 had "over 1 billion" transistors, while the A8 features 2 billion. As mentioned before, the "up to 84x faster" claim doesn't mean all that much given the comparison, but what is easier to understand is that the A8 has a 25% faster CPU and 50% faster GPU compared to the A7, according to Apple.
Beyond the manufacturing changes, Apple has added something called "sustained performance", because Apple says that traditionally, processors will lose performance over time. Phil Schiller illustrated this with a chart showing performance dipping over 20 minutes of use, but Apple forgot to label the Y-axis, so the chart doesn't really mean anything. However, with "sustained performance", the A8 will supposedly be able to run at full power without burning your hand, which would be very nice.
The game demoes that followed showed games that had up to 100 animated characters on screen at a time, with impressive frame rates, but textures and shading that isn't quite up to console level yet. Still quite impressive for a mobile device. Testing will show whether the new "sustained performance" will put a hit on battery life, but for now Apple is claiming battery life that is at least on par with the iPhone 5s.
img source: The Verge