The App Power is a score that calculates the overall visibility or weight of your app in the Store. It is a dynamic score that is directly influenced by the downloads an app drives or its category rankings. You can find more info on the App Power and interesting examples as to how it can be used right here.
The ASO score is calculated based on the extent to which your app metadata follows the ASO best practices. In the Overview section we give you an appreciation of each element of your app (Title, Subtitle, etc.) that goes from from red, yellow to green. Green means you are following the best ASO practices (for example, you are using all available characters in the subtitle field, or you have uploaded and app preview etc.). AppTweak then calculates the overall ASO score of your app and gives you a final grade which is the ASO score.
To improve your ASO Score, you can check the elements that aren't in green in the overview and see how to improve them, it's like a checklist. Improving the App Power is much more complex, since it means improving the overall quality of an app (more downloads, better conversion, better reviews etc.)
It is also important to keep in mind that an app can have an ASO score at 90% and still have a low app power. This is typical for smaller or new apps. An app can also have an app power at 100% and a very low ASO score, this is typical for highly popular apps that don't really need ASO to be found on the store such as Snapchat.
Check out this post for more details on the App Power