I am a Physicist who is learning Differential Geometry. I would like to know what are the advantages of intrinsic view of manifolds than extrinsic view?
This question can be considered to be in continuation to this question
What is the difference between intrinsic and extrinsic manifold?
In order to have an extrinsic view, you need to first have some ambient manifold to start with. In particular, you run immediately into trouble with complex manifolds since the only complex submanifolds of $\mathbb{C}^n$ that are of positive dimension(s) are noncompact (this follows from the maximum principle using the coordinate functions). If you are going to need to make an exception for $\mathbb{CP}^n$ then you might as well make it in general.
One reason why mathematics moved from considering everything concretely in the 19th century to abstractly in the 20th is that it separated out the concrete realization from the abstract notion, and in the process gave rise to interesting generalizations and new problems (e.g. Banach manifolds and infinite-dimensional Lie groups such as $\operatorname{Diff}(M)$, and questions such as the isometric embedding problem). Having said that, there are occasions you still want the extrinsic view because it is easier to picture (and in some case crucial, such as actually studying submanifolds of a given manifold).