Yes, the tsunami is a result of an earthquake while the storm surge is the pile up of water from high winds. I think I got that much right.
Otherwise, how are these physically different? Are they both just big walls of water that come crashing ashore, or is that the tsunami while the surge is much more gradual and longer lasting? Does the surge just follow the path of the hurricane, or does it radiate out in all directions and affect areas that are not otherwise affected by the hurricane? Is there any difference in the destructive power?
In a tsunami, the wave is the entire depth of the ocean while a storm surge (and other wind driven waves) are mearly on the oceans surface. The practical difference is that once the wind is no longer driving the storm waves, they rapidly loose energy. A tsunami, on the other hand, is a much bigger wave and thus doesn’t lose its energy.
When they approach land, a tsunami is a large wave, while a storm surge is a net rise in the surface of the ocean, somewhat akin to an extreme high tide.