Difference between Factory pattern and conditional Class?

I have some questions on the Factory pattern, I'll try to illustrate them using the example of handling HTTP errors in JS.

As far as I know, a "proper" factory would instance different ErrorXXX classes depending on the arguments passed to the Factory, like so:

    class HTTPError {
        createError(code) {
          if (code === 'NOT_FOUND') {
            return new Error404()
          } else if (code === 'NOT_AUTH') {
            return new Error401()
          }
    
          // etc....
    
        }
    }

But since in this example all of the classes will have the same properties you could have the conditional assigning values to those properties in a single class, instead of instantiating many different classes

    class HTTPError {
      createError(code) {
        let status = 500
        let message = 'Server error'

        if (code === 'NOT_FOUND') {
          status = 404
          message = 'Page not found'
        } else if (code === 'NOT_AUTH') {
          status = 401
          message = 'Resource not authorized'
        }

        // etc...

        return {status, message}
      }
    }

My questions are:

1. What are the different purposes of these two patterns?
2. Which one would be better suited for the task in the example?
3. Does the second pattern have a name?