Expanding just a touch on my comment:
One simple way to see that vectors and covectors are not as naively identifiable as your friend says is to realize that there are settings where the spaces they form aren't naturally isomorphic. They are developed independently on abstract manifolds, via the tangent and cotangent structures, respectively.
In particular, the only reason they are naturally isomorphic in Euclidean space is the presence of a natural Riemannian metric. It's this metric that gives you the sort structure you need to convert covectors (actually covector fields) into vectors (actually vector fields). This mapping is called the "musical isomorphism." You can get a similar mapping any time you have a non-degenerate two-form, e.g. a symplectic form would also work. But in the absence of such structure -- on an abstract finite-dimensional smooth manifold -- there is no such natural isomorphism, and you can't naturally identify the tangent and the cotangent structures (although any given tangent and cotangent space are isomorphic, by a dimensional argument).
This is a really important distinction, even in physics, as many have pointed out. It will show up when you learn how to integrate differential forms; you can't talk about "gradients" unless you have a Riemannian metric. It will also show up when you learn the symplectic structure of Hamiltonian mechanics: the natural setting for that is the cotangent bundle, but you can use the Legendre transform, sort of like the musical isomorphism, to push that physics into the tangent bundle, which gives you Lagrangian mechanics.
You might also be interested in the following discussion of how to "visualize" a covector, which can help to give an intuitive sense of why, though isomorphic, covectors and vectors are different things: