I assume you already know that MongoDB is a NoSQL database system which stores data in the form of BSON documents. Your question, however is about the packages for Node.js.
In terms of Node.js, mongodb is the native driver for interacting with a mongodb instance and mongoose is an Object modeling tool for MongoDB.
mongoose is built on top of the mongodb driver to provide programmers with a way to model their data.
EDIT: I do not want to comment on which is better, as this would make this answer opinionated. However I will list some advantages and disadvantages of using both approaches.
Using mongoose, a user can define the schema for the documents in a particular collection. It provides a lot of convenience in the creation and management of data in MongoDB. On the downside, learning mongoose can take some time, and has some limitations in handling schemas that are quite complex.
However, if your collection schema is unpredictable, or you want a Mongo-shell like experience inside Node.js, then go ahead and use the mongodb driver. It is the simplest to pick up. The downside here is that you will have to write larger amounts of code for validating the data, and the risk of errors is higher.