Good can be used to describe just about anything. Good hair. Good food. Good looks. Good house. Good feeling. Adroit is used typically used to describe a skill. An adroit actor. An adroit martial artist. As a previous comment mentioned, you will rarely use the word adroit.

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.

"Adroit" is a word we've taken directly from French, and even the definition of the word tells us the person's skill is expert, precise and quick. You would use this only in specific situations. "Good at ____" is more common and general.

Ya I've never heard that word before and I would guess many english speakers have never heard it either, so you don't need to remember it :P