Projective and proper are distinct notions

Under the finite-dimensional H-projective convention of Projective morphisms before Proj, projectivity strengthens properness: a projective morphism is a closed immersion into a relative projective space followed by the projection, and Projective morphisms are proper proves that every such morphism is proper. This is the implication that holds with no extra hypotheses; properness by itself is the weaker notion, defined by separatedness, finite type and universal closedness, with no reference to an ambient projective space (Proper morphisms).

The converse genuinely needs extra hypotheses, and over an algebraically closed field it is false. Closed gluing of two projective three-spaces is proper constructs a proper $k$-scheme by gluing two copies of ${\mathbb{P}}_k^3$ along a line and a smooth plane conic in each, exchanging the line and the conic under the identifications; the glued scheme is proper over $k$, but it admits no closed immersion into any ${\mathbb{P}}_k^N$. The obstruction is a degree argument. If such a closed immersion existed, then the pullback $\mathcal{L}$ of $\mathcal{O}(1)$ would restrict to $\mathcal{O}(n_1)$ and $\mathcal{O}(n_2)$ on the two copies of ${\mathbb{P}}_k^3$, with $n_1,n_2>0$, because on each copy the restriction is the pullback of $\mathcal{O}(1)$ along a closed immersion (Line bundles on projective three-space and their restrictions). Restricting instead to the glued curves, and using that a twist on ${\mathbb{P}}_k^1$ determines its index (The twist index on the projective line is an isomorphism invariant), the identification of the first line with the second conic gives $n_1=2n_2$, while the identification of the first conic with the second line gives $2n_1=n_2$. The two equations force $3n_1=0$, hence $n_1=n_2=0$ in $\mathbb{Z}$, contradicting $n_1>0$. So no such immersion exists.

Thus "projective" and "proper" are genuinely distinct: the first is a special case of the second, and the second does not imply the first. The companion examples page carries the coordinate model of the glued scheme and the full nonprojectivity calculation; the empty scheme, which is both proper and projective, is not the witness, and no Noetherian hypothesis is involved in either direction.