Hi Amanda, you can use the words "distinct" and "different" in both formal and informal situations actually. It's just that "distinct" is used less frequently.
"Distinct" can be used to mean "different", but it is a "stronger" word and stresses that 2 things are not just different from each other, but that they have recognizable differences which make them unique and of different kinds.
For instance... My house can be separated into 4 distinct spaces. vs. My house can be separated into 4 different spaces.
When "different" is used, it just means that they are not the same. But when you use "distinct", it tells me that each of these 4 spaces have characteristics or features which make them very different and unique from each other.