Are DBSCAN and dbscan from the sklearn.cluster package different?
Such questions are easily answered if you check the source code yourself.

https://github.com/scikit-learn/scikit-learn/blob/1495f69242646d239d89a5713982946b8ffcf9d9/sklearn/cluster/dbscan_.py#L350

One is an object-oriented wrapper with a more common API, the other the underlying function that does the actual work. Since the wrapper just calls the function, the results are not substantially different, only the presentation.

Hence, usually you won't need both.