Simple examples distinguishing conformal equivalence from projective equivalence?

Question

What are simple, easy-to-visualize examples of:

1. Two manifolds that are conformally equivalent but not projectively equivalent?

2. Two manifolds that are projectively equivalent but not conformally equivalent?

(I understand "conformal equivalence" to be, roughly, that there is a bijection that preserves angles; and "projective equivalence" to be, roughly, that there is a bijection that preserves geodesics.)