New Switch vs Old Switch - What's the real difference?

Is it just some firmware update that forces the screen to dim, thereby increasing the battery life?

No, this would barely be noticable.

The major difference is that the Tegra X1 (T210 "Logan") has been "die-shrinked", which means it is built on a smaller, more efficient process (in this case 16nm instead of 20nm) which improves efficiency leading to less power consumption and thus a longer battery life.

It's now called Tegra X1+ (or T214 "Mariko").

This guide has a pretty substantial comparison review between the older and newer models of the Switch.

Hardware

The newer Switch model does have a few upgrades to its hardware. Below is a chart comparing the models' internal components.

Improvements

• Display

  The new Nintendo Switch has a different display than the original model. The new display is slightly brighter and warmer (more red, less blue). The change in colors seems to be to make it more color accurate, though some gamers may prefer the blues of the original model. Several early reviewers noticed it with side-by-side comparisons. Kevin Kenson confirmed it with the help of a color expert.

• Battery

As you already noted, the newer model of the Switch is stated to have a better battery life. The guide ran some stress tests using different scenarios between the two models in the game Breath of the Wild.

After the stress test was finished, the newer Switch's battery was able to power the console longer than the older model. So while the battery model is still the same, the power consumption of the newer model is better.

It was also noted that the newer model takes less time to fully charge the Switch; this is likely related to the improvements to efficiency of the power consumption.

Other

While the newer model does have some upgrades to the hardware, the game load times, system audio, base internal storage, and size/weight of the Switch were not changed and compare equally to the older model of the Switch.

The guide did state they noticed download speeds were slightly faster, but this may also vary based on your internet speeds.