If you're not using MongoDB, the database, then there is no need to have any type of driver/odm solution. NodeJS will run just fine without these.
If you are attempting to connect to a MongoDB database instance, then you will need some sort of driver. I have never heard of a MongoJS, but maybe you are referring to the MongoDB driver? This driver will help you communicate with the database, such as saving documents or retrieving them.
MongooseJS is an object modeling library. It sits on top of the MongoDB driver and manages relationships and object mapping. It can detect changed properties in objects, and then run validation and update operations.
Long story short: You don't need to explicitly install both, as Mongoose will include the MongoDB driver when installing through NPM. You don't need either if you are not connecting to a mongodb database.