We can think of the Complex Numbers as an extension of the Real Numbers, similarly we can think of the Projective Plane naturally as a nice extension of the Euclidean Plane. But, when we go from real to complex numbers we lose some structure, for example total ordering, we $\textit{can't}$ ask about ordering anymore.
$\textbf{Question:}$ What do we lose by working in the Projective Plane? In other words, what "euclidean" questions don't make sense in the projective context?
You lose the vector space structure and even the affine structure and even the additive group structure when you go from $\mathbb C$ to $\mathbb P^1_\mathbb C$.
More concretely, given two points in $\mathbb P^1_\mathbb C$ it doesn't make sense to talk of their difference as a vector.
The general context is that $\mathbb C$ has both algebraic and geometric features while $\mathbb P^1_\mathbb C$ has no algebraic structure whatsoever but has many geometric structures:
As a topological space, manifold, Riemannian manifold, Riemann surface [ not synonymous at all with the former !], conformal space, algebraic curve,...