The main difference is in frequency of use.
We all use the phrase 'good at' millions of times in our lives. It's one of the commonest expressions in the language.
By contrast, I don't think I've ever said the word 'adroit' a single time in my entire life. A large proportion of native speakers probably don't even know what it means or how to pronounce it.
You can happily forget the word 'adroit' if you want to. Or if you don't, you should keep it in your passive vocabulary.